Serial No. 120 

DEPARTMENT OF COMMERCE 

U. S. COAST AND GEODETIC SURVEY 

E. LESTER JONES, SUPERINTENDENT 



PHYSICAL LAWS UNDERLYING THE SCALE 
OF A SOUNDING TUBE 



By 
WALTER D. LAMBERT 

Geodetic Computer 



Special Publication No. 61 




PRICE, 5 CENTS 

Sold only by the Superintendent of Documents, Government Printing Officei 

Washington, D. C. 




WASHINGTON 

GOVERNMENT PRINTING OFFICE 

1920 



Monograph 



Serial No. 120 

DEPARTMENT OF COMMERCE 

U. S. COAST AND GEODETIC SURVEY 

w 

E. LESTER JONES, SUPERINTENDENT 



PHYSICAL LAWS UNDERLYING THE SCALE 
OF A SOUNDING TUBE 



By 
WALTER D. LAMBERT 

Geodetic Computer 



Special Publication No. 61 




PRICE, 5 CENTS 

Sold only by the Superintendent of Documents, Government Printing Office, 

Washington, D. C. 



WASHINGTON 

GOVERNMENT PRINTING OFFICE 

1920 







PREFATORY NOTE. 



The use of the sounding tube, or depth recorder, has long been 
recognized as a convenient and rapid method for getting approxi- 
mate soundings in depths of 100 fathoms or less. It has never been 
considered an instrument for accurate surveying, and, indeed, the 
depths shown by two tubes of different patterns thrown overboard 
at the same point, or even by two tubes of the same pattern, have 
often exhibited surprising discrepancies. 1 Moreover, it does not 
appear that the method used in graduating tubes in current use has 
ever been published in detail. It is the purpose of this paper: 
(1) To provide as correct a scale as possible for the sounding tubes of 
the new Coast and Geodetic Survey type, the scale to be computed 
on assumptions definitely stated and by a stated formula; (2) to 
provide a method of correcting the depths as read directly from the 
scale for variations in temperature and in atmospheric pressure, so 
that the tube may be used in surveying as an instrument the precision 
of which is comparable with that of the lead; (3) to provide a com- 
pilation of physical data likely to prove convenient in a further 
study of the subject. 

This paper was nearly ready for publication two years ago, when 
the war delayed its completion. In the meantime, the first draft of 
the manuscript was read by the late Dr. K. A. Harris, of this Survey, 
and several suggestions were derived from his memoranda. Many 
of the tables were prepared with the help of the Section of Tides and 
Currents. 

1 The more extreme discrepancies are due partly to accidents in the working of the tubes, partly to 
irregularities in their bore. 

2 

D, of J* 



N» 



CONTENTS 



Page. 

Prefatory note 2 

General statement 5 

General principles of the sounding tube 6 

Approximate formulas 7 

More accurate formulas and corrections: 

Departure from perfect gas 8 

Vapor pressure of water 9 

Correction of the reading of the tube for departure from assumed normal 

conditions 11 

Amount of compressed air dissolved in the water 13 

Absorption by the walls of the tube 17 

Change in volume of the trapped water 18 

Correction for variation in the density of the water and for the acceleration 

of gravity 20 

Correction for fluctuations of the water surface 21 

Situation of the point whose depth is recorded 21 

Numerical data and construction of the tables 23 

Summary of conclusions 25 

Note 1. Rate of absorption of air by water 26 

Note 2. Cer-tain questions in mechanics connected with the sounding tube 28 

Note 3. The heat evolved by the compression of the air in the tube 29 

Note 4. Compressibility of sea water 32 

Schedule of mathematical notation: 

English characters 33 

Greek characters 35 

Examples in the use of the tables 35 

Tables: 

Table 1. Miscellaneous physical data 37 

Table 2. Density of sea water 38 

Table 3. Vapor pressure of sea water 39 

Table 4. Compressibility of sea water 40 

Table 5. Absorption of atmospheric gases by sea water 40 

Table 6. Pressure of sea water at various depths 41 

Table 7. Table for scale of sounding tube 42 

Table 8. Special table for scale of Coast and Geodetic Survey tube 43 

Table 9. Corrections to sounding tube readings for temperature and pressure. 45 

Table 10. Effect of absorption of air on the scale of a sounding tube. 45 

ILLUSTRATIONS. 

Figure 1. — Diagram showing position of point where depth is recorded 22 

Figure 2. — Diagram illustrating absorption of gas by a liquid 26 

3 



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PHYSICAL LAWS UNDERLYING THE SCALE OF A 

SOUNDING TUBE. 



By Walter D. Lambert, Geodetic Computer. 



GENERAL STATEMENT. 

The idea of measuring the depth of water by the pressure it exerts 
is one that must have occurred to inventors, even in early times. 
It has been embodied in many forms of apparatus, 1 none of which, 
however, were ever widely and continuously used until Lord Kelvin 
(then Sir William Thomson) put his depth recorder on the market 
in 1871. Improvements in detail have since been made, but the fun- 
damental principle remains the same. The Tanner-Blish tube, now 
used by the United States Navy, was patented in 1899. The main 
advantage of these contrivances is that the sounding can be taken 
while the ship is under way. 

The volume of air follows, approximately at least, Boyle's law 
that the volume is inversely proportional to the pressure, and there- 
fore at great depths and pressures the volume is so small and varies 
so little in absolute amount, even with considerable change of depth, 
that the scale divisions become very minute, and, therefore, this type 
of instrument has seldom been used for depths over 100 fathoms. 
It is believed that by increasing the size of the instrument the sound- 
ing tubes of the United States Coast and Geodetic Survey type could 
be made useful up to 250 fathoms. 

To avoid the difficulty mentioned above, various devices have been 
invented, among them the following: 

(1) Apparatus similar in principle to an aneroid barometer. 

(2) Apparatus for measuring pressure by the change in volume of 
sea water itself or of some other fluid. 

(3) Apparatus in which the gas to be compressed has an initial 
pressure of several times the prevailing atmospheric pressure. 

(4) Apparatus in which, instead of measuring the variable volume 
of air under compression, a constant volume of more or less com- 
pressed air is allowed to expand till the pressure is reduced to the 
prevailing atmospheric pressure. The expanded volume is a 
measure of the original compression. 

None of these forms of apparatus has attained any wide use among 
navigators or hydrographers, and in this paper only those sounding 
tubes are considered in which the pressure of the water is exerted 
against air initially at atmospheric pressure. 

1 Ericsson invented in 1836 a device similar to Sir William Thomson's. 

5 



6 U. S. COAST AND GEODETIC SURVEY. 

GENERAL PRINCIPLES OF THE SOUNDING TUBE. 

All sounding tubes are alike in fundamental conception, which is 
that of balancing the pressure of water against the pressure of com- 
pressed air. The tube is full of air of the same constitution, tempera- 
ture, and pressure as the air near the surface of the water. It is 
inclosed in a protective metal case in which it is dropped to the bottom 
where the depth is to be determined. The tube being open at the 
lower end, the pressure of the water compresses the air in the tube 
and the water enters to occupy space left as the air recedes under 
compression; from the relative amounts of air and water in the 
tube the water pressure, and therefore the depth, may be deduced. 

The fundamental differences between tubes of different makes lie 
in the method of ascertaining the amount of water in the tube. There 
are tubes in which the water rising in the tube leaves a record of the 
maximum height reached by it (which corresponds to the lowest 
depth). In one type the inner surface of the tube is of ground glass, 
which is wetted by the rising water and remains wet when the water 
falls in the tube as the latter is raised to the surface again. It has 
been found that the wetted area sometimes spreads upward a little 
in the tube after the water has withdrawn. This effect is probably 
due to capillary action in the roughened surface of the tube. In 
another type of tube the inner surface is coated with a substance 
that changes color when touched by sea water, so that the latter 
leaves an automatic record of the height to which it has risen. In 
the new United States Coast and Geodetic Survey tube the opening in 
the tube is closed by a valve and spring, the resistance of the latter 
being equal to the pressure of 1 fathom of water. When the depth of 
the tube is greater than 1 fathom, the water pressure overcomes the 
resistance of the spring behind the valve and the water enters and 
continues to enter until the tube has reached its lowest point. When 
the tube is pulled up again, the pressure outside the tube is less than 
the air pressure within, and the valve remains closed, thus trapping 
the water. The water is thus brought to the surface in the tube, and 
its amount may be measured in any convenient way. The way pro- 
vided for ordinary surveying purposes is by means of a graduated 
rod of uniform diameter. The rod, which is of known dimensions, is 
plunged into the tube until the water in it is about to overflow. The 
depth to which the rod must be inserted gives the amount of air space 
in the tube, and thus the depth. The depth may be read directly from 
the scale marked on the rod, or by marking the point to be measured 
with a sliding marker provided for the purpose, and laying the rod 
along a suitably graduated scale. For accurate experimental work 
the water could be measured by weighing it with care. 

This paper has been prepared with the new Coast and Geodetic 
Survey tube especially in view. Most of the considerations are, how- 
ever, of a general nature and will apply to any tube in which the 



SCALE OF A SOUNDING TUBE. 7 

volume of atmospheric air is used as a measure of the water pressure. 
Whenever the point under discussion does not concern sounding tubes 
of older types, attention is drawn to the fact. 

APPROXIMATE FORMULAS. 

To graduate the scale of a sounding tube, a relation must be found 
between the volume of the air in the tube under compression and the 
corresponding depth. A fair approximation to the desired relation 
is readily obtained. Assume that the air at the surface exerts a 
pressure 1 Pi and has the absolute temperature t v As the tube de- 
scends this air is compressed by the water pressure, and the tempera- 
ture is thereby raised. However, the whole apparatus will quickly 
take on the temperature of the surrounding water, 2 and we shall 
assume that the compressed air in the tube has, in fact, the tempera- 
ture of the water at the point whose depth is to be measured. Let 
this temperature be t 2 on the absolute scale and let 8 be the mean 
density of the water between the bottom and surface, and let p 2 be 
the pressure due to the weight of the water. 

Then 

p 2 =--Ugh. (1) 

In this equation g is the acceleration of gravity, and 1c is a constant 
dependent on the units used. Suppose that the air obeys the laws 
of a perfect gas; then at the surface 

p.v^Rt^or, v^-- 1 - (2) 

In this equation R is the gas constant, its value depending on the 
units used. At the depth li, where the total pressure is p t +p 2 an d 
the temperature t 2 , the corresponding volume v 2 is given by 

m^_ m ( 

The ratio of the volume occupied by the compressed air to the original 
volume of the tube is, therefore, 3 

v i *t Pi + ltty~h 

J For brevity in this publication the word pressure is used in the sense of intensity of pressure or force 
per unit area. 

2 For the contrary case see note 3, p. 29. 

3 The following reasoning may make the point clearer: Suppose for the moment that R is so chosen and 

that v is the volume in cubic centimeters occupied by 1 gram of air under standard conditions (temperature 

O °C, pressure 1 atmosphere). If the pressure be made p\ and the temperature h, each gram will occupy vi 

V 
cubic centimeters, and the tube containing V cubic centimeters will contain — grams of air under these 

conditions. The air in the tube being now subjected to pressure p\ + p° and temperature ti, each gram 

V Vi 

occupies volume vi cubic centimeters, or —grams of the compressed air in the tube will occupy V— cubic 

t'l Vi 

Vo Vo 

centimeters. — is, therefore, the fraction of the total volume that is occupied by the air. 1— — is the 

Vi Vi 

fraction of the volume occupied by the sea water. 



8 U. S. COAST AND GEODETIC SURVEY. 

This simple calculation omits several rather obvious considerations : 
(1) At depths still within the range of the instrument the behavior 
of the air departs perceptibly from that of a perfect gas; (2) vapor 
tension of sea water has not been allowed for; (3) the fact has not 
been considered that the water in the tube may dissolve some of the 
compressed air in the tube, thus diminishing its apparent volume. 

MORE ACCURATE FORMULAS AND CORRECTIONS. 

DEPARTURE FROM PERFECT GAS. 

Within a range wider than is here needed the behavior of air under 
different conditions of temperature ft), pressure (p), and volume (t>) 
may be represented by Van der Waals's equation: 

(p+fy(v-b) = Rt. (5) 

In this equation a and b are quantities small in comparison with 
p and v. The numerical values of a and b depend on the gas under 
consideration and the units used in measuring p and v. In deriving 
equation (5) Van der Waals assigned the following physical interpre- 
tations to a and b : b represents four times the volume of all molecules 

concerned, and -5 is the pressure due to the mutual attraction of the 

molecules, which Van der Waals takes as inversely proportional to 
the square of the volume of the gas, so that a is the factor of propor- 
tionality. 

To square with the observed facts over wide ranges of temperature 
and pressure, a and b must be made variable, but for the compara- 
tively limited ranges involved in the present case it is quite sufficient 
to take them as constant. In the computation of the tables the 
values of a and b used were deduced such as to represent the results 
of physical experiment over about the range of conditions to which 
the normal sounding tube would be subject; that is, Van der Waals's 
equation was used as an empirical formula. The experimental data 
were taken from Winkelmann's Handbuch der Physik, volume 1, 
part 2, page 1259, and the values deduced are on p. 25 of this paper. 
Equation (5) by a simple transformation gives the value of p when v 
is known, but for our purposes it is better to have v given in terms of 
p. This would require the solution of a cubic equation; but by 
taking advantage of the smallness of a and b, which are taken as 
small quantities of the first order, equation (5) may be transformed 
into the following approximate form, which retains small quantities 
of the second order. The equation is readily derived by the method 
of successive approximations. 

Rt . , a T 2ab a 2 ~| ,_ N 

v= J +h -m + lim-(myf- (6) 



SCALE OF A SOUNDING TUBE. 9 

The term in square brackets, even when multiplied by p, is nearly 
always negligible in our calculation, except when extreme refinement 
is required at great depths. 

VAPOR PRESSURE OF WATER. 

Equation (5) or (6) applies to dry air only. The pressure of the 
water vapor is added to the pressure of the air and is the same as if 
the latter were absent. Let us define the quantity jh more strictly 
and let it signify the pressure at the surface due to the air only 
and let us call p\ the pressure of the water vapor in the atmosphere 
immediately above the surface, be that what it may. At sea 
it will probably be close to the tension of saturated vapor cor- 
responding to the temperature of the air. When the air is com- 
pressed by water pressure, the water vapor is made to occupy a 
smaller space, and the air undoubtedly becomes over-saturated 
and the vapor condenses, the condensed moisture being added 
to the sea water in the tube. Even if the air in the tube were 
initially quite dr} T , it would take up moisture from the sea water in 
the tube and become saturated. The water given up by the air, or 
taken up by it, replaces or is replaced by an equal amount of water 
from outside of the tube, but in any case the amount is very small. 
The tension of saturated vapor over a salt solution is somewhat 
lower than it is over pure water. (For numerical values, see p. 39.) 
The pressure on the water may also affect the saturation tension of 
the water vapor above it, but only by a very minute amount. 

The pressure to be used in equation (5) or (6) for computing the 
volume under conditions at the surface is p l only. Let p r denote 
the pressure of saturated vapor above water corresponding to the 
salinity and temperature of the sea water in question. The pressure 
that compresses the air is : 

Total atmospheric pressure -I- pressure due to weight of water = 
Pi+p\+p 2 ) which is equal to the counterpressure exerted by the air 
and by the saturated vapor: that is, it equals p 2 +p' ', so that the air 
pressure by itself without the vapor tension, which is to be used in 
equation (6) to determine the volume of the compressed air, is p = 
Pi+p\+p 2 — p'- Using the more exact relation (6) in the same 
way as equation (1) was used to obtain equation (4), we have, 
instead of (4), the more exact relation 

v, \p 1 +p\+p 2 -p' RtjV'Kp, BtJ°) 

= ilp 1 +p\+p 2 -p'~\IW 2 ~M) Pl T L 1 VW 2 ~sd^J (7) 

With an assumed set of conditions of temperature, etc., and with 
the depth as the only variable, the formula (7) is readily adapted for 
tabulation, being less complicated than it looks. 
152573°— 20 2 



10 U. S. COAST AND GEODETIC SURVEY. 



It may be written 
in which 



v 2 _ C x 



^1 V2 + <?2 



C t , (8) 



«-| i f_a__ Pl ±\ ' f (9) 

C 2 =Pi+P'i-p', do) 

**(-* l -\ v 

(y3 ~ 1 (_a h\ ' ui; 

l ~\(Rt 1 r RtJ Pi 

The C's, though of comphcated form, may be computed once for 
all for the given set of assumed conditions. It should be noted 
that equations (8) to (11) do not take into account the small term 

(j>*\2 ~ /pf\3 m equation (6). For depths less than 110 fathoms the 

effect of this term is only a few units of the fifth place of decimals, 
and may be allowed for by making C 3 slightly variable. The correc- 



v 
tion term to — is 

+ 
or 



f 2ab a 2 1/ p 2 + C 2 \ 

l(Rt 2 r w 2 yj\ v, J 
+ [w^(^] ( ^ +Q f' (lla) 

This term is to be added to the expression (8) for —, or may be 

allowed for by combining it with <7 3 . The quantity p 2 is nearly 
proportional to the depth. The small correction terms proportional 
to the square of the depth are worked out later. (See p. 24.) 

When there is a valve spring of known strength, as in the tubes of 
the Coast and Geodetic Survey pattern, its effect may be allowed 
for by noticing that it acts like the vapor tension of water (p f ) in 
reducing the resistance that must be sustained by the compressed 
air alone. To allow for the spring, formula (10) may be rewritten 

2 = p 1 +p l , -p f s, (lib) 

in which s is the pressure of the spring expressed in the same units 
as the other quantities. A method practically equivalent to the 
foregoing is to compute the table using (10) in its original form, but 
taking p 2 in (8) for the depth 7i — n instead of for depth h, and to 
tabulate the results under depth 7i, in which n denotes the number 
of units in the depth of water exerting a pressure equal to the 
resistance of the spring. In the tubes of the Coast and Geodetic 
Survey pattern n = 1 fathom. 



SCALE OF A SOUNDING TUBE. 11 

CORRECTION OF THE READING OF THE TUBE FOR DEPARTURE FROM 

ASSUMED NORMAL CONDITIONS. 

When the conditions vary much from those assumed and it is 
desired to make accurate allowance for that fact, as would be the 
case in experimental work, it will probably be most satisfactory to 
compute directly from formulas (7) to (11a). However, for the degree 
of accuracy required under ordinary working conditions, differential 
formulas suffice. These may be derived from the simpler basic 
formula (4) instead of from the more complex formulas (8) to (11). 
The effect of the omitted terms would seldom amount to more than 
0.5 fathom for depths less than 100 fathoms. For completeness, 
however, the formula derived from the basic formulas (8) to (11) is 
also given. 

From (4), 

lo g Gr) =1 °g *2~log *!+log Pt-log (p t +Jc8gh) 
Differentiate this expression, keeping — constant. 

<7^ dt x dp x dpi+ledgdh 
*2 ^i Pi Pi + kdgh ' 

which solved for dh gives 

The quantity t^- is the height of a column of water exerting a 

pressure p x and is about 5J fathoms, since p x is always about 1 
atmosphere. 

In deriving (12), dt l7 dh 2 , dt 1} and dp x were treated as infinitesimals. 
The corresponding finite increments Ah, At 2 , etc., may be substituted 

for them without serious error so long as -r~i -—> etc., are small. 

In accordance with a well-known principle, increased accuracy will 
be obtained by adding to each original quantity half the correspond- 
ing increment; that is, by replacing h by 1i-\--k" > t 2 by 2 2 + ~o~ 2 ' e ^ c - 
In this way (12) becomes 



Equation (13) gives a second approximation after an approximate 
value of Ah has been found by any method. 



^VEY. 
12 U. S. COAST AND GEODETIC SURVEY. 

A simplified form which will give this approximate value with 
sufficient accuracy, and may usually be used to give the definitive 
correction, is obtained by omitting the increments on the right-hand 
side; and by replacing both t 2 and t x by T, equal to a mean between 

them(V=^- 2 ). 
This gives 

M= T( 7fc+ S) (A<2_A<l)+?l ^ ! ' (14) 

In the Coast and Geodetic Survey tube the standard conditions are: 

^ = 60° F. = 519° F. above absolute zero. 
* 2 = 50° F. = 509° F. above absolute zero. 
2? t = pressure of 30 inches of mercury. 

Therefore, T=514°, and (14) becomes 

Ah = 0.001946 [h +5.46] [t ± -t 2 - 10°] +0.0333 ABxh. (15) 

h and Ah are in fathoms and 30 + AB is the reading of the barometer 
in inches. From (15) Table 9, page 45, has been computed. The 
above formula is for any tube the scale. of which is based on the 
assumptions stated. In strictness for the Coast and Geodetic Survey 
tube h should be replaced by h — 1, since the valve spring takes up 
the pressure due to 1 fathom of depth. 

The more exact formula deduced from equations (8) to (11) is 
stated below without proof. The proof is similar to that of (12), 
but is tedious owing to the length of the formulas. The formula 
reads 

2 L 12 i J i L \ 1 — Z{p x / 

-%^2 + G^ d f\^~^-^(v 2 + C 2 y-p^\-d{p\- V '). (16) 

As before, the differentials may be replaced by the corresponding 
finite increments, and each finite quantity increased by half its incre- 
ment, as in (13). 

In deriving (16) some terms, in which a or b occurs multiplied by 
other quantities likewise small, have been dropped. The symbol 

z x stands for p 2 2 ~ pt- Water pressure p 2 may be converted into 
depth by noting that 

p 2 = 0.18141945/*, + 0.000000780A 2 . 



SCALE OF A SOUNDING TUBE. 13 

(See eq. (35), p. 24.) It is seen that p 2 + C 2 in (16) cone- 

v 
sponds to h+T, It m (12), so that an estimate may be made of the 

effect of the additional small terms in (16). As a numerical 
example, take p 2 + C 2 = 20 atmospheres, corresponding to about 



105 fathoms depth 
dp, 



1 
>t?v> corresponding to dt x 



> 27° C., also 



1 
> yq ana 



Pi 



1 



oTv , corresponding to one inch difference in the 
barometer reading. The additional terms are (1) n£r* n ' m 

the correction for t 2 , (2) the factors 1 + WtS and 1 — g 1 p 1 , which in the 

approximate formulas are each replaced by unity, and (3) the terms 
in C 3 in the corrections for t x and Pu which together with the vapor 
pressure terms d QV ' —p') are omitted in the approximate formulas. 

The terms in -pj] 1 . p affects the correction for t 2 by about 1/20 

L Z 1 

of its amount in the extreme case supposed or by nearly half a fathom. 
The factor 1 + p 2 j- 2 iorp 2 + C 2 = 20 atmospheres, affects the correction 

for air temperature by less than j^ of its amount, or less than 



0.03 fathom. The terms in C 3 amount to about tt^ of the total 



1 

100 

corrections for air temperature and air pressure, or about 0.1 fathom 
and 0.04 fathom, respectively. The factor l—2 1 p 1 has less effect than 

the factor 1 + p^rV The effect of vapor-pressure terms dip/ — p') 

is practically independent of the depth. The vapor pressures 
are functions of the temperatures of the air and water, respec- 
tively, but do not vary proportionally to the temperature over any 
considerable range, and so have not been included with the tempera- 
ture terms. An inspection of Table 3 will show that Px'ovp' , particu- 
larly the former, might vary by 20 mm. of mercury = Wo atmos- 
phere corresponding to 0.14 fathom. 

AMOUNT OF COMPRESSED AIR DISSOLVED IN THE WATER. 

It is difficult to make a satisfactory allowance for the amount of 
compressed air dissolved in the water, because in practice there is 
never time for the process of solution to be completed. In order, 
however, to ascertain the maximum possible correction due to this 
cause, we shall work out the effect of absorption of air by water 
when continued till the latter is saturated. 

*The vertical bars indicate numerical value regardless of algebraic sign. 



14 U. S. COAST AND GEODETIC SURVEY. 

At a given temperature the mass of gas absorbed per unit volume 
of water is, by Henry's law, proportional to the pressure of the gas; 
and since the density of the gas is very nearly inversely proportional 
to its pressure, the volume of gas absorbed per unit volume of water, 
or the ratio between the volume of the gas absorbed and the volume 
of the absorbing water, is, within wide limits, independent of the 
pressure of the gas. At ordinary temperatures water will absorb 
about 2 per cent of its bulk of air. (See p. 40 for more precise data.) 

When there is no shaking together of air and water, the process of 
saturating the water with the air is a decidedly slow one. (See 
note 1, p. 26.) Just how much this process would be hastened by 
the motion and jarring of the sounding tube incidental to taking a 
sounding, it would be impossible to predict from theory. The cor- 
rection to be applied in practice could be determined by experiment 
only for a rough average for ordinary working conditions. Table 
10 (p. 45) shows the limiting values (1) when no absorption of air has 
taken place and (2) when the absorption of gas is complete. The 
scale to be used in practice would be very nearly that of the first 
limiting case for small depths and perhaps less near for larger depths 
for which the graduation is unfortunately more sensitive. 

The two scales on sounding tubes in general use are the Thomson 
scale and the Parmenter scale. The former scale is used on the tubes 
designed by Sir William Thomson (Lord Kelvin) and the second on 
the Tanner-Blish tubes made in the United States by D. Ballauf . No 
statement of the formula by which the Thomson scales were origi- 
nally made has been found. One of the scales was measured and the 
measurements examined to see whether any allowance was made for 
absorption, but apparently there was none, or, if any was made, it was 
very small. Lieut. Parmenter of the U. S. Navy, who computed the 
scales for the Ballauf tubes, has left an unpublished memorandum 
of his method. Unfortunately it is rather hard to follow. Appar- 
ently, however, allowance was made for absorption. His statement 
is "That at 100 fathoms the absorption of air was 0.13 per cent, and 
at 5 fathoms the absorption was 0.04 per cent." It is not clear 
whether 0.13 per cent of the original air space is intended or 0.13 per 
cent of the volume of water in the tube, the latter mode of statement 
being in accordance with our usual conception of the physical phenom- 
enon. A study of the figures of his table would seem to indicate 
that 13 per cent (not 0.13 per cent) of the air space itself may be 
meant. This would be about 0.13 inch at 100 fathoms. This allow- 
ance for absorption seems somewhat excessive. Tests of the new 
United States Coast and Geodetic Survey tubes under various pres- 
sures were made at the Bureau of Standards, and the results of 
these tests were examined to discover the amounts of absorption. 



SCALE OF A SOUNDING TUBE. 15 

The tests, however, were not made with this phenomenon especially 
in view, and nothing definite could be learned from them except 
that, even at pressures equivalent to 100 fathoms of water, the effect 
of absorption on the size of the air space is probably less than 1 
millimeter (0.04 inch), even when considerable time is allowed. The 
tubes, however, were not used under working conditions, but simply 
put into the testing apparatus and subjected to the required pres- 
sure. There was none of the moving or jarring incidental to the 
usual process of taking the sounding. 

A careful preliminary test of the new Coast and Geodetic Survey 
tubes made by the Coast and Geodetic Survey in Florida Straits in 
May, 1919, indicates that a scale computed on the assumption that 
absorption may be neglected gives good results, even at depths of 
100 fathoms and more. 

Lieut. Parmenter's statement of the amount of the absorption is 
given as the result of 36 tests on the U. S. S. Prairie in 1900. Details 
of these tests are not given, so they can not be examined to see if the 
apparently large absorption can be otherwise accounted for. How- 
ever, it may be that Lieut. Parmenter's table is substantially correct 
on this point. Further experiment on this point is desirable. 

The method by which the table for the second limiting case, that of 
complete absorption, was computed is as follows: 

We may assume that the water, being continuously subject to the 
air pressure p 1 of about 1 atmosphere, has already absorbed the 
corresponding amount of gas. This assumption is found to be 
approximately correct. (See Krummel, Handbuch der Ozeanogra- 
phie, vol. 1, p. 296.) Under compression some of the air is absorbed 
into the water, which would tend to reduce the pressure; but the 
latter is maintained by the inflow of water to take the place of the 
air absorbed, and this process continues until equilibrium is attained. 
The earlier forms of sounding tube communicated freely with the 
water outside without any check valve intervening; and if unlimited 
time were allowed, there was no reason why all the air might not be 
absorbed into the water of the tube and ultimately be diffused into 
the surrounding ocean. In the United States Coast and Geodetic 
Survey tube this diffusion into the outside water is prevented by the 
valve. Even with this form of apparatus, the volume of air might 
be so small that in time the water in the tube alone would absorb all 
the air. This can not happen, however, at depths of less than 250 
or 300 fathoms, and for such depths the tube is not used. 

In computing the volume of air remaining after absorption is 
complete it will be assumed that Henry's law is exact and that the 
density of the air is the same as for an ideal gas. These approxima- 
tions are amply accurate for the purpose in hand, since they are in 



16 U. S. COAST AND GEODETIC SURVEY. 

themselves near the truth and are used merely to calculate a small 
correction, and not the principal quantity. Let p 2 be the density due 
to the pressure of Pt + p 2 . Let m be the number of grains of air 
absorbed by 1 cubic unit of water under unit pressure, and let Av be 
the diminution in volume of air due to absorption, so that v— Av is the 
reduced volume. Let w be the volume of water when equilibrium 
has been established. The water already contains the amount of 
air due to the atmospheric pressure p 1} so that the additional mass of 
air absorbed is that due to the additional pressure p 2 and is given by 
the equation 

p 2 Av = wmp 2 (17) 

and since the water flows in to replace the absorbed air, 

v— Av + w=V, (18) 

in which V is the entire volume of the tube. 

Let pj be the density of the gas at pressure p lf then 

/^'a±a. . (19) 

Pi Vi 
Then from (17), (18), and (19), by eliminating w and p 2 and solving 
for Av, we get 

(V—v) mp 2 



Av- 



(£+')'■- 



mp 2 

or, 

(V-v)^p 2 
Av= Pl 



Vi+V2-~7 1 V2 



rnpi 
Pi 



The quantity — — is a constant for a given temperature and is equal 
Pi 

to the fraction of its volume of air that water will absorb. Put — — 

Pi 
equal to a; then 

(V— v)a 



Av = 



i + ;H_ a eo) 



V2 

The numerical value of a is about 0.02. 

It may be noted that the oxygen and nitrogen of the air are not 
absorbed into the water in the same proportion in which these gases 
exist in the atmosphere, the proportion of nitrogen in the dissolved 
air being less than that in the atmospheric air. The undissolved 
air is, therefore, of slightly different composition from ordinary 
atmospheric air, and in strictness its physical constants would be 
slightly different. This effect is obviously too minute to need further 
consideration. 



i 



SCALE OF A SOUNDING TUBE. 17 

The constant a in equation (20) is 0.01942 and is derived as follows: 

From Table 5 on page 40 it appears that for a salinity 32.5 parts per 

thousand and a temperature of 10° C, 1 liter of sea water will absorb 

6.51 c. c. of oxygen and 12.22 of nitrogen, or 18.73 c. c. in all. This 

figure refers to the volume at 0°. To reduce the volume to 10° with 

283 
sufficient accuracy for the purpose in hand, multiply it by ^5* which 

is the ratio of the absolute temperatures. The result is 19.42 c. c. 
per liter; that is, sea water at 10° will absorb 0.01942 of its own volume 
of air, or about 2 per cent, as previously stated. 

The effect of the air dissolved in the water on the vapor tension 
of the latter is also negligible. (See J. J. Thomson, Application of 
Dynamics to Physics and Chemistry, p. 173.) The effect of the 
dissolved air on the volume of the dissolving water is likewise negli- 
gible. The increase in volume of the water is about equal to the 
volume that the air would occupy under a pressure of 2,500 atmos- 
pheres. (See Winkelmann, Handbuch der Physik, vol. 1, pt. 2, 
p. 1521.) 

In the tubes of the new Coast and Geodetic Survey pattern there 
is a quasi absorption that can be easily guarded against. In measur- 
ing the amount of water contained in a tube it i ; s inverted and the 
air bubble rises through the w T ater. In this way minute air bubbles 
may be formed and remain in the water for some little time, thus 
increasing its apparent volume. To remove the bubbles the water 
may be thoroughly stirred with a fine wire. 

ABSORPTION BY THE WALLS OF THE TUBE. 

There is still to be considered the possibility that, under the pres- 
sure up to 20 atmospheres, the water or compressed air might perme- 
ate the walls of the tube to an extent sufficient to vitiate the measure- 
ments. This possibility is to be feared much more in the case of 
brass than of glass, which has long been used in accurate physical 
experiments at high pressure. With reference to brass or bronze 
the United States Bureau of Standards furnishes the following 
information. 

The rate of air or water passing through brass or bronze would depend upon many 
f actors f of which the following might be named : Composition of material ; condition 
of material, cast, rolled, forged, etc.; temperature; pressure; depth immersion; etc. 
The bureau has been unable to find any specific data on the rate of flow of air or water 
hrough brasses or bronzes. 

Carpenter and Edwards (Proceedings Institution of Mechanical Engineers, 1910, 
p. 1597) state that from their investigations, a pure copper-aluminum bronze contain- 
ing from 9 to 11 per cent of aluminum has the best ability to withstand high pressures 
(14 to 20 tons per square inch). This material when properly cast did not leak until 
just before rupture; the original article gives the necessary precautions which should 
be taken in casting. This alloy is also only very slightly attacked by fresh or salt 
water. 

152573°— 20 3 



18 U. S. COAST AND GEODETIC SURVEY. 

The possibilities of error from this source would be diminished by 
giving as little time for the pressure to act as may be consistent with 
other requirements. The experience of the Coast and Geodetic 
Survey does not indicate that the error arising from the permeability 
of the tube is serious, but no data have been obtained as to the exact 
amount or the rate of permeation. 

CHANGE IN VOLUME OF THE TRAPPED WATER. 

There is one particular in which an accurate calculation for a tube 
of the Coast and Geodetic Survey type differs from the calculation 
for a tube of the recording type mentioned on page 6. In the latter 
type the height of the water in the tube is recorded automatically 
in situ; in the Coast and Geodetic Survey tube the volume of the 
water is measured at the surface and under surface conditions of 
temperature and pressure (but see p. 20), whereas we are concerned 
with its volume under the conditions that prevailed at the depth 
measured. As on page 16 let w denote the volume of the water at 
the given depth, v the volume of air in the tube, and V the entire 
volume of the tube. Call Aw the total increment of w due to change 
of temperature and decrease of pressure experienced in going from 
the given depth to the surface and put 

Aw = A x w + A 2 w, (21) 

in which A x w is the increment due to change in temperature and A 2 w 
the increment due to the decrease of pressure. These increments 
are so small that they may be computed independently. Since 
v + w= V= constant, Aw = —Av. 

Let d 1 and 8 2 denote, respectively, the densities of the water at air 
temperature t x and water temperature t 2 . Then, since the mass of 
the water is unchanged by the change in temperature, 

wb 2 = (w + A 2 w)d l 

or A 2 w = w ^ 2 ~ dl) - (22) 

Since the 5's are nearly unity, we may put in rough calculations 

A 2 w = w(S 2 -\). (23) 

We have further, 

A t w = Wfxp 2 , 

ix denoting the coefficient of compressibility for the temperature and 
salinity of the water, and p 2 the water pressure. The vapor pres- 
sures J?/ and p' are much too small to need consideration. (See 



SCALE OF A SOUNDING TUBE. 19 

Table 3, p. 39.) Since — (formula (8), p. 10) is the ratio of the volume 

occupied by the compressed air to the total volume of the tube, we 
have 



v=^ 2 Fand 



Therefore, 

Aw = A 1 w + A 2 w = w ^1' 2 + ^^ V(l -^)(m2> 2 + ^^)- (24) 
The corresponding correction to 7i, All, is found by 

*- _ A ,^ =Aw ,[r|(|)]=^(^ + A^). (25) 

dJi\vJ 

The minus sign is used before Av because we are reducing back 
from the volumes at the surface to the volumes under water, whereas 
the A's have been denned as the changes caused by going in the op- 
posite direction. w( — j is essentially negative. Ah may be sepa- 
rated into Ajt+Aji. 
where 

l-J 

M" d/vX *"?*' < 2G ) 

dh\vj 

which is the the correction due to change in pressure, and 

-j ^2 

^STSft*^ (27) 

bh\vj 

which is the correction due to change in temperature. 

A^ has been allowed for in computing Table 8 for the scale of the 
Coast and Geodetic Survey tube. Its value is small, being less than 
2 fathoms for a depth of 100 fathoms. A 2 h has not been allowed for 
in computing Table 8, as a further correction of the same sort has to 
be introduced when the actual temperatures differ from the standard 
assumed temperatures, and the introduction of part of the necessary 
correction into the table itself would complicate rather than simplify 

matters. The quantity -^ — - is a function of the temperature, but 



20 U. S. COAST AND GEODETIC SURVEY. 

is not a linear one even approximately, so that this term does not 
lend itself to combination with the other temperature corrections in 

i_2s 

equations (13) or (16). The quantity -\ / \ * s tabulated for each 

dJiyvJ 



depth on pages 42-44. Although .the expression for kt ( — ) 

T»o«rlil"ir n^ rtarhipprl frnm fimiatinna (R\ anr] f90^ in n?9/>i.iftfl it. ia r 



can 



readily be deduced from equations (8) and (20), in practice it is more 

convenient to get its numerical value from the differences in the 

v 
tabulated values of — by some one of the formulas connecting finite 

differences and derivatives. 

Under certain circumstances it may not be necessary to make the 
correction for change in volume of water due to change in temper- 
ature. If the difference in temperature is not great and the meas- 
urement of volume is made very promptly, the water may be as- 
sumed to have maintained its original temperature. If this assump- 
tion is made, it will be advisable to keep the graduated rods that 
are introduced to measure the volume at somewhere near water 
temperature. On the other hand, if the correction is to be applied, 
sufficient time should be allowed for the water to take on the tem- 
perature of the air. 

CORRECTION FOR VARIATION IN THE DENSITY OF THE WATER AND FOR 
THE ACCELERATION OF GRAVITY. 

Tables 7 and 8 were computed with a standard surface density of 
water equal to 1.025 and a standard acceleration of gravity (# 45 ) 
equal to theoretical gravity at sea level in latitude 45°. When act- 
ual conditions depart much from the assumed conditions, as when 
soundings are taken in fresh or nearly fresh water, a correction must 
be applied. 

The water pressure is proportional to the product (density) X- (ac- 
celeration of gravity) X (depth), or, with the notation previously 
used, 

p 2 = UgJi. (28) 

For a given reading of the tube scale, p 2 is constant, although each 
of its factors may vary. Logarithmic differentiation of (28) gives 



=-(M> 



By substituting finite increments for differentials, and denoting by 
A 3 h the correction for difference between the actual and standard 
values of density and gravity, we get 



SCALE OF A SOUNDING TUBE. 21 

AS is to be taken so that 1.025 + A<5 shall represent the mean density 
from the surface down to depth h for the actual distribution in depth 
of salinity and water temperature and a pressure of 1 atmosphere; 
A5 can be deduced from Table 2 for the assumed conditions. No 
allowance is to be made for increase in the density of the water due 
to the pressure of the water above it, since this effect is small and 
has already been allowed for with sufficient accuracy (p. 24). The 

term — may be dropped, except in computations of unusual refine- 
rs 

ment. In such cases take 

^i= -0.0026 cos 2<p. (30) 

In (30) <p is the geographic latitude. Tables of — or related 

945 

quantities are commonly given in collections of meteorological 
tables for the purpose of reducing the height of the barometer to 
latitude 45°. 

CORRECTION FOR FLUCTUATIONS OF THE WATER SURFACE. 

In finding the depth to which the sounding tube has been sub- 
merged, no correction need or should be made for the fact that the 
crest of a wave has passed over the spot where the tube lies, and that 
its maximum depth below the instantaneous surface is therefore 
greater than the depth below the mean surface of the water, which 
is the depth commonly desired. This statement applies to surface 
waves such, as are raised by the wind, and whose length (distance 
from crest to crest, or from trough to trough) is less than the depth 
of the water, and whose amplitude is small compared with the depth ; 
for it is well known that the oscillatory effects of such* waves die out 
very quickly as the depth below the surface increases, and that the 
water pressure at the bottom under the trough, or under the crest is the 
same, being equal to the static pressure due to the depth below the 
undisturbed mean surface. The statement does not apply to long 
waves like tidal waves, "tidal wave" being used in its proper sense 
of a periodic wave produced by the attraction of the sun or moon. 
Depths found by the sounding tube are to be corrected for the stage 
of the tide in the same way as are soundings taken with the lead. 

SITUATION OF THE POINT WHOSE DEPTH IS RECORDED. 

In making an accurate comparison of the depth shown by the 
tube with that shown by the lead at the same point, two points 
must be remembered: (1) As the tube sounding is usually made, 
the tube may not go to the bottom, but the protective metal case 
is attached to a weighted rod several feet from the end. This is 
done in order to prevent the case and tube from being injured by 
striking the bottom. The rod remains upright after striking 






\ 



22 



U. S. COAST AND GEODETIC SURVEY. 



bottom, the weight being at the lower end, and a constant allowance 
must be made for the distance between the end of the rod and the 
lower end of the tube. (2) Equilibrium between the air pressure 
within and the water pressure without is established, not at a fixed 
point in the tube, but at the boundary surface between the air and 
water in the tube. 1 

This would require a variable correction the amount of which 
could be easily deduced from Table 7 or 8, column 2, pages 42-44, 
and from the dimensions of the tube. To find the exact point where 
equilibrium would be established requires the solution of a quadratic 
equation even when Boyle's law is assumed to be exact. If a de- 
notes the length of the tube, b the height of a column of water exert- 
ing a pressure equal to that of the atmosphere, li the depth of the 
bottom of the tube, 2 x the height to which the water rises in the tube, 
then 



a 



b + 7i — x 



a — x 



■t or x= 



a + b + h 



^(•±|±»y^ A (3D 





Fig. 1. — Diagram showing position of point where depth is recorded. 

This is the equation frequently given for graduating the tube. In 
the Coast Survey tube the correction for distance between end of 
tube and water surface, which is precisely the quantity x, is (to the 
nearest tenth of a fathom) 0.3 fathom for depths of 16 fathoms and 
over. The correction due to the position of the tube on the weighted 
rod is found by direct measurement. 

i The fact that in the Coast and Geodetic Survey tube there is a spring valve that takes up the pressure 
due to 1 fathom of water does not affect the correctness of this statement. 

2 The quantity h elsewhere denotes the depth of the boundary surface between the air space and the 
water in the tube. 



SCALE OF A SOUNDING TUBE. 23 

NUMERICAL DATA AND CONSTRUCTION OF THE TABLES. 

The tables for the physical properties of sea water are based 
principally on the data and tables in Krummel, Handbuch der 
Ozeanographie, second edition. The table for the density of water 
of different temperatures and degrees of salinity is constructed as 
follows: Knudsen's empirical formula, cited by Krummel (vol. 1, 
p. 237) for the density of water at 0° C. 1 and different degrees of 
salinity was first used, and then the densities at other temperatures 
were interpolated from the table on pages 232 and 233 of KrummePs 
work. The vapor pressure for different temperatures and degrees of 
salinity were computed indirectly from the formulas and tables on 
pages 241 and 242. The tables give the effect of the salinity of the 
water on the boiling point. The alteration in the boiling point was 
converted into the equivalent alteration of vapor pressure by means 
of the Smithsonian Physical Tables (sixth revised edition, 1914). 
The ratio of the vapor pressure of water of given salinity to the vapor 
pressure of pure water, as deduced from a consideration of their 
boiling points, was assumed to be independent of the temperature. 
(See Thomson, Application of Dynamics to Physics and Chemistry, 
pp. 175-177 ; also Chwolson, Traite de Physique, Vol. Ill, p. 956-969.) 
The vapor pressure of pure water was taken from the Smithsonian 
Tables. In this way Table 3, page 39, was prepared. 

In constructing the tables of water pressure corresponding to a 
given depth, the density of the sea water was taken as 1.025. For 
a temperature of 10° C. this corresponds to a salinity of 32.5 parts 
per thousand. The assumed density seems to represent fairly well 
conditions in American coastal waters. The density of the sea 
water is as a rule a trifle higher elsewhere than the value here adopted, 
but the difference is not important in practical work. 

The compressive force exerted by the water is proportional to the 
acceleration of gravity. This quantity varies between the Equator 
and poles by about one two-hundred ths part of itself. In accurate 
investigations of the scale of the tube the appropriate local gravity 
should be used, but for ordinary purposes we may use the normal 
acceleration for sea level in latitude 45°, which has been taken as 
980.62 centimeters-per-second per second. The density of mercury 
at 0° C. is 13.595945 (Landoldt and Bornstein, 4th ed., p. 45). One 
standard atmosphere, which is the pressure of 76 centimeters of mer- 
cury under the above gravity, is therefore 76x980.62x13.595945 = 
1,013,210 dynes per square centimeter. One fathom = 182.8804 
centimeters. Therefore, the weight of each fathom of sea water exerts 
a pressure of 0.18141945 standard atmosphere. This is the quantity 
~kbg in equation (4), if Ji be in fathoms. There are slight corrections 

i a- = -0.093+0.81495-0.0004825 2 +0.00000685' 3 . For <r see p. 33. 5 is salinity in parts per thousand, by 
weight. 



24 U. S. COAST AND GEODETIC SURVEY. 

to this figure owing to the increase in density of water under the 
compression of the water above it and the increase in gravity as the 
center of the earth is approached. These effects are small, but, 
being easily evaluated, have been included in making up the tables. 
Let 8 = 5 + A5 and g = g + 8g be the values of density and gravity, 
respectively, at any depth ft, 8 Q and g Q being their values at the sur- 
face. If p be the pressure and 1c a constant depending on the units 
used, then 

dp = Tcdgdh = Tc(g + Ag) (5 C + Ad)dJi = Tc (8 g + 8oAg + g<A5)dh. (32) 

In (23) the small term of the second order in Ag A8, has been omitted. 

1 d h 
If jS be the coefficient of compressibility, by definition tj- = ^; or, 

approximately neglecting the variation in 5, A8 = (38 p; and with the 
approximate value p = 7c8 g h, we have 

A8=pJc8 2 gJi. (33) 

In approaching the center of the earth through free air by ft meters, 
the acceleration of gravity increases 0.0003086 ft centimeters per sec- 
ond per second. 1 In this case, however, the approach toward the center 
is not through free air but through water, the attraction of which, if it 
be treated as an indefinitely extended layer of thickness ft, is 2irf8Ji, in 
which/ is the gravitation constant 667.3 X 10~ 10 C. G. S. units. Since, 
when the point is below this layer of water, the attraction of the 
water is reversed in direction, double the above effect must be sub- 
tracted from the rate of increase of gravity in free air. 

The numerical expression of 2irf8ji is 0.0000429 ft; therefore the 
increase of gravity in approaching the center of the earth through ft 
meters of water is 0.0002228 ft centimeters per second per second, or 
A# = 0.000407ft, when ft is in fathoms instead of in meters. For 
brevity write Ag = c7i, c standing for the above numerical value 
0.000407. Resuming equation (32) we find for the corrected value 
of the increment of water pressure, 

dp 2 = (^5 ^o + Tc8 ch + plc 2 d 2 g 2 h)dh, (34) 

whence 

ch 2 2 7k 2 

p 2 = 7c8 gji + (&5 £ ) —a + jS (k8 g ) ^- • 
9o & 

The quantity /3 = 451xl0~ 7 (see p. 40); therefore, on substituting 
numerical values, 

p 2 = 0. 181419457i + 0.0000000377T* 2 + 0.000000742ft 2 , (35) 

or 

p 2 = 0. 18141945ft + 0.0000007807t 2 . 

The unit of p is the standard atmosphere and of ft is the fathom. 

1 Investigations of Gravity and Isostasy. Spec. Pub. No. 40, U. S. Coast and Geodetic Survey. See p. 93. 



SCALE OF A. SOUNDING TUBE. 25 

The constants of Van dor Waals's equation (5) arc taken as follows. 

a = 0.00246, 6 = 0.00102, R = 0.0036650 = 1/272. 85. 1 
Absolute zero is taken at — 273° C. The unit of v is the volume 
of the mass of air which will just fill the chamber of the tube under 
standard conditions; namely, a pressure of one atmosphere and a 
temperature of 0° centigrade. The above numbers were deduced by 
trial to fit approximately the data given in Winkelmann, Handbuch 
der Physik (vol. 1, pt. 2, p. 1259). They are intended to apply to 
pressures between 1 and 30 atmospheres and temperatures between 

0° and 100° C. 

SUMMARY OF CONCLUSIONS. 

To get the best results from a sounding tube, allowance must be 
made for the temperature of the air and of the water and for the 
barometric pressure; also for the fact that air does not exactly con- 
form to Boyle's law and for other possible sources of error. The 
tables for the scale of the tube given in this work contain these allow- 
ances and corrections as far as it was practicable to include them. 
Other tables given are intended to facilitate the calculation of the 
necessary corrections. 

If the sounding tube is carefully made and used and the correc- 
tions for temperature, etc., are properly applied, the accuracy of the 
results should be practically the same as that of soundings carefully 
taken with a lead. The present tubes are available up to depths of 
100 fathoms. It is believed that a larger tube could be made that 
would give satisfactory results up to 250 fathoms. 

The advantages of a tube over the lead are the rapidity with 
which the sounding can be made and the fact that it is unnecessary 
to stop the vessel in order to get a good sounding. The sounding 
tube gives the mean depth of the water unaffected by fluctuations 
of level due to short surface waves. 

In taking a sounding sufficient time should be allowed for equi- 
librium to be established, both as to the inflow of water and as to 
the equality of temperature of t'.e tube and of the surrounding water. 
On the other hand, the time must not be unduly prolonged so as to 
give time for the air to diffuse into the water or for the air and water 
to permeate the walls of the tube. Care should be taken to avoid 
the quasi absorption (p. 17) in the Coast and Geodetic Survey tube. 
In measuring the volume of the air space above the water brought 
to the surface in a Coast and Geodetic Survey tube, conditions of 
time and temperature should be such that the temperature of the 
water in the tube may be definitely assumed to be either that of the 
water at the depth where the sounding was taken or else that of the 
surrounding air. (See p. 20.) In the latter case, the correction for 
change in volume of water with change in temperature should be 
applied. 

1 Since a and b enter the equation, R is no longer exactly^* as for a perfect gas; otherwise p and v would 
not be unity together. 



26 



U. S. COAST AND GEODETIC SURVEY. 



Further study is desirable on the absorption of air by water under 
working conditions and of the permeation of the walls of the tube 
by air and water. It would be desirable also to consider whether 
the reading of the tube is affected by the kinetic pressure due to its 
descent through the water or by the shock of its striking the bottom. 

NOTE 1.— RATE OF ABSORPTION OF AIR BY WATER. 

Let a chamber of gas be maintained at a given temperature and at 
constant pressure against a column of liquid of uniform cross section 
capable of absorbing the gas into solution. Let this column be closed 
at the end opposite the gas chamber, and let the length of the column 
be denoted by I, and let x denote the distance of any cross section of 
liquid from the surface of separation of liquid and gas. Let p be the 
density of the gas dissolved in the liquid at this cross section at the 
time t. 

The partial differential equation which p must satisfy is 




X 



k. _ 



(pas 



Liquid 




— = Jc—> 
dt dx 2 



(36) 



Fig. 2. — Diagram illustrat- 
ing abscrplion of gas by a 
liquid. 



(See Winkelmann, Handbuch der Physik, vol. 
1, pt. 2, p. 1446.) The quantity Jc is a con- 
stant, the coefficient of diffusion. It is re- 
quired to build up a solution of this equation 
P=f(x, t), in which f(x, t) must satisfy the 
following conditions: 

(1) f(x } °° ) =p s ; in which p 8 is the gas density 
in the liquid when the latter is saturated. 

(2)f(x, 0) = except when x = 0, in which 
£ case f(x, t) is indeterminate. 

when x = l for all values of t 7 

since the further end of the tube is closed and 
there is no flow of gas across it. 

To simplify the printing of exponential quan- 
tities with complicated exponents we shall use 
the notation exp (z) for e z , z standing for any 
expression simple or complicated and e being 
the base of the natural logarithms. 



*> U -0 



A simple particular solution of (36) is 

p = exp( — 7ccH)sm ex (37) 

or p = exp ( — Jcc 2 t) cos ex 

In (37) c is an arbitrary constant. If we choose the sine function 
and put e = -sr, where n is an odd integer, we satisfy condition (3) ; and 



SCALE OF A SOUNDING TUBE. 27 

by making use of the Fourier expansion, 

. f. sin y sin 5y , e e . , OQ> . 

1=- sin yH — g-M =- ° (38) 

we can build up a solution of (36) that satisfies the require* 1 condi- 
tions; namely, 



1 /-25K 2 A . 57rz "I) 

+ 5 ex pv— if-; sm_ 2r J) 



(39) 



The mean gas density over the whole column of liquid p m is defined 

i r z 

Pm = T I P<fo (40) 

Therefore, at any time, t, 

P m -P s {l-^Lex P ^ 2 -j + -exp^^^j + ^exp^-^^-|.j](41) 

According to this equation p m reduces to zero when £ = 0, as it should, 
since "o~ = 1 + q + 9^ + Jq > also when t = co , p m becomes p 3 . 

Winkelmann gives the following numerical values of k in c. g. s. 
units for temperatures of 16° C. (Handbuch der Physik, vol. 1, pt. 2, 
p. 1450.) 

Coefficient 
of diffusion 
Gas. k. 

C0 2 0.0000159 

N 0000200 

0000187 

N 2 0000156 

CI 0000127 

NH 3 . . : 0000128 

For a numerical example let us take Z = 50 centimeters, which, is 
rather shorter than the column of liquid in the sounding tube when 
the latter is used at a considerable depth. Let us consider also the 
case of nitrogen, 'which has a larger coefficient of diffusion than any 
other gas in the preceding table. Accordingly, 7t: = 2xlO~ 5 . In 

JiTT 2 t 

order for -j™- to be of such size that p m may approximate even roughly 

5 X 10 7 
to its limiting value p s , i must be large. Take t = » — seconds, or 

7T" 

over 58 days; by computing from (41) we find that, even with this 
large value of t, p m is less than one-fourth (23 per cent) of the possible 
density of saturation. 



28 U. S. COAST AND GEODETIC SURVEY. 

The values of the coefficients of diffusion given in the preceding 
table are for pure water. Although they appear to be somewhat 
larger for sea water, absorption, whether of fresh water or of sea- 
water, when effected by diffusion alone, is an extremely slow process. 

In sea water, under natural conditions, absorption is aided by 
convection currents due to differences of temperature or to differ- 
ences of salinity produced by evaporation. It is also supposed that 
particles of dust act as nuclei of condensation and as carriers for 
minute quantities of gas, and so hasten absorption at lower depths. 
It has been suggested also that there are nuclei of condensation of a 
electrolytic nature present in sea water which act as carriers and may 
serve to account for the observed difference in the rate of diffusion 
between air into sea water and air into pure water. 1 

In the sounding tube absorption would be hastened by all the 
influences just mentioned, and probably much more by the motion 
of the water itself in entering the tube, and by the other motions and 
concussions incidental to the process of taking a sounding. The in- 
formation available leads us to conclude that if the correction for the 
quantity of air absorbed be omitted the resulting error in depth will 
be small, and that it will be better not to attempt to make the cor- 
rection until further tests have been made. 

NOTE 2.— CERTAIN QUESTIONS IN MECHANICS CONNECTED WITH THE 

SOUNDING TUBE. 

The problem has been treated hitherto as a statical problem; that 
is, as if either the tube descended to the bottom with extreme slow- 
ness or else as if the valve were not released to admit the water until 
the tube had reached its lowest point, and as if, furthermore, the 
water lost all momentum immediately after passing the valve 
opening. 

It is found that the tube will descend to a depth of about 105 
fathoms in about 45 seconds. A freely falling body would cover this 
distance in about 6 seconds. The buoyant effect of water will dimin- 
ish the apparent acceleration of gravity and increase the time of 
descent, but not greatly. Most of the difference between 45 and 6 
seconds is to be explained by the resistance of the water. This resist- 
ance implies a pressure which would be additional to the statical 
pressure of the water. Formulas for the motion of a body subject to 
a constant acceleration and to a resistance proportional to the 
square of the velocity (Newton's assumption, which is probably nearly 
correct) may be found in many works on mechanics. 2 

1 For an account of diffusion under natural conditions and of experiments dealing with the rate of diffu- 
sion of gases through liquids, also references to the literature of the subject, see Krummel, Handbuch der 
Ozeanographie (vol. 1, p. 298). 

2 Some idea of the total resistance encountered may be derived from the following general considerations. 
If a body of mass to in a resisting medium like water is acted on by a constant force like gravity and by the 
resistance of the medium, which is some function/ (v) of the velocity v, it being understood that resistance 



SCALE OF A SOUNDING TUBE. 29 

In this case, however, the acceleration is not constant, since water 
is continually entering the tuhe, thus diminishing the buoyant effect 
of the water outside. The problem is thus rendered very complicated 
and requires for a numerical solution further experimental data, 
especially on the rate at which water under a given pressure would 
pass through the valve opening. 

It might be found that the opening was so large that, owing to the 
pressure arising from the motion, an amount of water would enter 
the tube in excess of the amount determined for the depth. If the 
valve opening is too small, it will require an excessive time to estab- 
lish equilibrium between the pressures inside and outside the tube 
On this account care should be taken not to raise the tube too soon 
to the surface. The matter seems to be one for experimental 
investigation. 

When the tube reaches the bottom, it is traveling with considerable 
velocity and may be stopped with more or less suddenness, depending 
on the nature of the bottom. The first effect of the shock would be 
to throw the water in the tube against the valve at the bottom, thus 
preventing the entrance of any more water. Possibly when the 
water in the tube rebounds from the lower end there might be a 
chance for more water to enter, but, if the valve opening is small, 
this does not seem likely. Much would also depend on how nearty 
instantaneous is the adjustment of the amount of water in the tube 
to the depth. If the opening is small, this adjustment would require 
time, and it would be well to investigate the effect of letting the 
tube remain at the depth to be sounded for a longer or shorter 
period. 

NOTE 3.— THE HEAT EVOLVED BY THE COMPRESSION OF THE AIR IN 

THE TUBE. 

In all formulas for the volume occupied by the compressed air it 
has been assumed that sufficient time had been allowed for the air 
to take on the temperature of the surrounding water. As the tube 
in practice is lowered and quickly raised again, it may be of interest 
to estimate how much heat must be given out by the tube in this 
short time in order that our assumption may be justified. 

The amount of heat evolved in compressing a gas depends on how 
the compression is brought about. We shall calculate the amounts 
on three simple suppositions. The formulas used will be found in 
almost any elementary book on thermodynamics, or may be readily 

increases with velocity, then the velocity -will increase more and more slowly as a certain limiting velocity 
V, called the terminal velocity, is approached. The resistance/O) depends, among other things, on the 
size and shape of the body. When the acceleration is practically zero as the terminal velocity is ap- 
proached, terminal resistance/( V)=mg, g being the acceleration of gravity. This gives the limit to which 
the resistance or total pressure on the body approximates. What would be the pressure intensity on a 
particular point— for instance, on the valve— can not be estimated, even approximately, without a knowl- 
edge of the form of body. 



30 U. S. COAST AND GEODETIC SURVEY. 

deduced from the ones there given. The air is treated as a perfect 
gas, an assumption which greatly simplifies the formulas and gives 
more than sufficient accuracy for the purpose in hand, which is 
illustrative only. 

Case 1. — The compression is so gradual that the heat of compression 
is absorbed by the water as fast as it is involved and the air is all 
the while at the temperature of the surrounding water. This is 
called isothermal compression. 

Denote by J the mechanical equivalent of heat, by W the amount 
of work done compressing the gas isothermally at temperature t 1} 
from volume v t and pressure p x to volume v 2 and pressure p 2 , and 
H' the heat given out in the process. 

Then 

„, W Rt t , p 2 p x v x , p 2 p t v t , v t , -: 

In this case the heat given out is less than will be given out on any 
other admissible supposition. 

Case #.— Suppose the air to be surrounded by matter impervious 
to heat, so that heat of compression is retained. Suppose that the 
air retains its heat till the pressure p 2 is attained, and that the non- 
conducting layer is then removed. The temperature of the air, and 
the corresponding amount of heat and final volume attained under 
pressure p 2 will depend on how the pressure is applied. If the pres- 
sure is increased gradually, so that the volume and temperature are 
at every instant adjusted to the pressure, the compression is called 
adiabatic. If t" 2 and v" 2 denote the absolute temperature and the 
volume corresponding to pressure p 2 and if & denote the ratio of 
the specific heat of the air at constant pressure C p to its specific heat 

C 
at constant volume, C YJ that is, Jc=jf- the formulas for adiabatic 



compression are 



* 2 " = U° 



/ V 



l Y (44) 

2/ 



When the nonconducting layer is removed, the air cook at constant 
pressure p 2 to the temperature t t of the surrounding water and in so 
doing shrinks from volume v 2 " to v 2 , the volume it finally attained 
in Case 1. 

In so doing the heat II" given out is expressed by 

R" = C»{t 2 "-t>> 

-A^ft)"- 1 ] (45) 



SCALE OF A SOUNDING TUBE. 31 

Case 3. — In this case, as in Case 2, the heat is to he retained by a 
nonconducting layer, but the pressure, instead of being increased 
gradually, is suddenly increased from jh to p 2 . When the volume 
v 2 " f and the temperature t 2 rn corresponding to pressure p 2 nav e 
been reached, the nonconducting layer is removed, the air cools to 
the temperature t 1} and shrinks to the volume v x of Case 1. The 
formulas for II'" , the heat given out, are 

H" f = C p [t 2 "' — t 1 ] 

-jCPi-A) ' (48) 

As a numerical illustration we take Pi—1 atmosphere, p 2 = 20 
atmospheres, corresponding to a water pressure of 105 fathoms 
nearly, and ^ = 273° absolute = 0° C. If the C. G. S. system of units 
be used, we must take the pressure in dynes (1 atmosphere = 1.0132 
X 10 6 dynes), and the unit volume will be that of a gram of air under 
standard conditions = 773.5 c. c. We have further, by experiment, 
<7 P = 0.238, and 



fc=g=1.4; 



T 1.0132 X10 6 X 773.5 . Qf)v1n7 ,,_. 

J= 97Qvn9Qg/ T~\ = 4 - 22 x 10 ergs- ( 49 ) 



and from theory, 

: 273X 0.238 



a) 



The values of II will be expressed in gram calories per unit volume 
of air, and, to find the heat given out by each cubic centimeter of air 
at atmospheric pressure, we must divide the respective values of II 
by 773.5. 

Case 1. (Isothermal compression. Compression gradual, heat given 
out as fast as produced.) Each cubic centimeter of air at atmos- 
pheric pressure gives out enough heat to raise the temperature of 1 
gram of pure water by 0.°072 C. 

Case 2. (Adiabatic compression. Compression gradual, heat is 
retained till pressure of 20 atmospheres is reached and is then given 
out.) The temperature of the air is raised by 369.°5, but each cubic 
centimeter of air at the original atmospheric pressure gives out only 
enough heat in cooling to raise the temperature of one gram of pure 
water by 0.°114 C. 



32 U. S. COAST AND GEODETIC SURVEY. 

Case 3. (Compression sudden; heat is retained till equilibrium 
is reached at a pressure of 20 atmospheres; heat then given off.) 
The temperature of the compressed air is raised by 1482°, but each 
cubic centimeter of air at the orignal volume gives out in cooling 
only enough heat to raise the temperature of 1 gram of pure water 
by 0.°456 C. 

Sea water has a specific heat slightly less than pure water (0. 936 
for water of salinity 32.5 parts per thousand, density 1.026 at 0°) ; 
and if instead of reckoning by grams of pure water we reckon the 
heating effect in a cubic centimeter of pure water the figures in 
Cases 1, 2, and 3 will be, respectively, 0.°077, 0.°122, and 0.°487. 

The actual heating effect evidently lies between Case 1 and Case 3, 
probably much nearer to the former. In any event, it seems plain 
that, in spite of the limited time spent in taking a sounding, the air 
can take on practically the temperature of the surrounding water, 
as has been assumed. 

NOTE 4.— COMPRESSIBILITY OF SEA WATER. 

The mean coefficient of compressibility /* of a substance between 
the pressures P and P+p is defined by the equation 

H = 50 

In this equation v and v are the volumes corresponding to pres- 
sures P and P+p, respectively. The true coefficient of compressi- 
bility (ju ) at pressure P is the limit of the mean coefficient as p 
approaches zero, or 

1 dv d n . ,_ 1S 

"° = -^ = -^ (l °s v) - (51) 

V D 

We have also, since — =— , the p's being densities corresponding 
to the v% 

Mo= | (logp) =i|. ; (52 ) 

The most thorough investigation on the compressibility of sea 
water is by W. V. Ekman, in paper No. 43 of the "Publications de 
Circonstance" of the "Conseil Permanent International pour l'ex- 
ploration de la mer" entitled, "Die Zusammendrueckbarkeit des 
Meerwassers." His final result in his own notation is: 

l0V= 1 +0 4 000186p ~ [227+28 - 33< ~ - 551P + - 004<3]+ TTOO [105 - 5 
-0.87« + 0.02* 2 )]+(^^Y[4.5-0.U-^(1.8-0.060]. (53) 



SCALE OF A SOUNDING TUBE. 33 

In this equation fi represents the mean compressibility between 
atmospheric pressure and p additional units of pressure, so that when 
p is zero the pressure is 1 atmosphere. The unit of p is the bar, that 
is, 1,000,000 dynes per square centimeter. 1 One standard atmos- 
phere = 1.01323 bare, or 1 bar = 0.98694 atmosphere; t represents the 
temperature in degrees centigrade; c is a quantity connected with 
the density 5 of the water at 0° C, in such a way that density at 

°° c - 1+ i&r 

On the basis of formula (53) Tables 4a and 4b have been calculated. 
It seemed more convenient, however, to tabulate the true compressi- 
bility for p = 0, to make the unit of pressure the atmosphere instead 
of the bar and to use as one of the arguments the salinity instead of 
the density at 0°C. This has accordingly been done. 

In strictness the tabulated true conpressibility, /z , applies only 
when p = 0; practically, it may be taken for the mean compressi- 
bility for all pressures within the range of present sounding tubes. 
To take into account, however, the terms of formula (53) that do 
not appear in ju , we may proceed as follows: Suppose the change 
in relative volumes to be expanded in a series of powers of the incre- 
ment of pressure, p. From (50) and (51), the first term is evidently 
PIjl ; that is, 

~-^-=VoP+a 2 p 2 +a*P* , (54) 

the a's being coefficients independent of p, but for sea water dependent 
on the temperature and salinity. A brief table of the values of a 2 
is given on page 40. The unit of pressure is the atmosphere. a 3 is 
very small, indeed; according to (53), it is, when reduced to the 

atmosphere as unit of pressure, ( 1.76 — r-^— jxlO -12 . If it is desired 

to compute the change in relative density, the expression to be used 
is 

^^-° = MoP + (a 2 +Mo 2 )^ 2 (55) 

Po 
SCHEDULE OF MATHEMATICAL NOTATION. 

ENGLISH CHARACTERS. 

a Constant of Van der Waals's equation; definition, p. 8; numerical 

value, p. 25. 
a In equation on p. 22 only, special meaning. 
a 2 Coefficient in formula (55), p. 33, for change of volume under pressure. 

Numerical values in table, p. 40. 
a 3 See p. 33. 
6 Constant of Van der Waals's equation, p. 8, numerical value, p. 25. 

1 This unit is also called the megnbar, the bar being then defined as a pressure of one dyne per square 
centimeter. 



34 U. S. COAST AND GEODETIC SURVEY. 

b In equation on p. 22 only, special meaning. 

B Used only in AB, p. 12. 

c Special meaning, p. 24; different special meaning in note 1, p. 26. 
Cj, C 2 , C 3 Denned by equations (9), (10), (11), and (lib), p. 10. 

Cp, C Y Specific heats at constant pressure and constant volume, respectively, 
in note 3, p. 29. 

/ Used only on p. 24, and defined there. 

g Acceleration of gravity in general. 

g Q Acceleration of gravity at sea level. 

h Depth to which tube is submerged below sea surface; more specifically 
depth of bottom of air space, see p. 22. Working unit of h, the fathom. 
H' y H", H"' Quantities of heat, note 3, p. 29 only. 

J Mechanical equivalent of heat, note 3, p. 29 only. 

h General meaning, coefficient of proportionality between depth and 
water pressure; special meaning in note 1, p. 26, coefficient of diffu- 
sion of gas into liquid; second special meaning, note 3, p. 29 only, is 

a. 

l In note 1, p. 26 only, 
m Mass in general, note 2, p. 28; elsewhere number of grams of air absorbed 

by a cubic unit of water. 
n See p. 10, following equation (lib). 

p Pressure in general. Working unit of pressure the atmosphere. 
P and p Indicate particular pressures defined in note 4, p. 32. 

p x Atmospheric pressure at sea surface, exclusive of pressure due to water 

vapor, also a somewhat different meaning in note 3, p. 29. 
p\ Pressure at the surface due to water vapor. 

p / Pressure due to water vapor in the air space of the tube, and assumed 
to be the vapor pressure of saturation for water of the salinity and 
temperature of that surrounding the tube. 
p 2 Pressure due to the weight of the water, and also a somewhat different 

special meaning in note 3, p. 29. 
R The gas constant of Charles's law or of Van der Waals's equation. 
s See equation (lib), p. 10. 

S Salinity of water, see Table 1, pts. 1 and 2, p. 37. 
t Time in note 1, p. 26 only, elsewhere temperature. In the formulas 

temperatures are on the absolute scale unless otherwise stated. 
^ Temperature of the air, always on absolute scale in formulas. 
t 2 Temperature of the water, always on absolute scale in formulas. 
t 2 /// , t 2 " Temperatures defined in note 3, p. 29. 
T Defined, p. 12. 
v To suggest volume in general, and in particular volume of air in tube 

as on p. 16, except in note 2, p. 28, where v is used for velocity. 
v Initial volume under compression. Note 4, p. 32. 
v l Volume of unit mass of air under pressure p x and temperature t x ; denotes 

a different special volume in note 3, p. 29. 
v 2 Volume of unit mass of air when tube is submerged ; denotes a different 
special volume in note 3, p. 29. 
v/, v 2 '\ v 2 '" Volumes defined note 3, p. 29. 

V Entire volume of the tube except in note 2, p. 28, where V denotes 

terminal velocity. 
w Volume of water in the tube. 
W Mechanical work equivalent to W . 
x Two distances, defined where they occur, pp. 22 and 26. 
z x Defined, p. 12. 



SCALE OF A SOUNDING TUBE. 35 



GREEK CHARACTERS. 

a Volume of gas which unit volume of water will absorb. 

/3 Coefficient of compressibility of sea water. 
5 Density of sea water in general. 

5 Density of sea water at the surface. 

5 1 Density of sea water at air temperature and atmospheric pressure. 

5 2 Density of sea water at water temperal ure and al mospheric pressure. 
A Finite increment of quantity following, which see. 

A p A.,, A 3 Special finite increments, see pp. 18-20. 

e Base of natural logarithms = 2.71828 

n Mean coefficient of compressibility of sea water. See note 4, p. 32. 
n True coefficient of compressibility of sea water, at atmospheric pressure, 
see note 4, p. 32. 
p Denotes density in general in note 4; elsewhere a gas density, particu- 
larly in note 1, p. 26. 
p lt p 2 Densities of air due to different pressures, p. 16. 
Pm> P 8 See note 1, p. 26. 

a Defined in note 4, p. 32. 
<p Geographic latitude, p. 38. 

EXAMPLES IN THE USE OF THE TABLES. 

Tables 1, 3, 5, 9, and 10 are self-explanatory. The following 
examples may illustrate the use of the other tables: 

Example 1. — The volume of a quantity of water is 100 c. c. at 
atmospheric pressure, temperature being 5° C. and salinity 30. 
What will be its volume at the same temperature under an additional 
pressure of 20 atmospheres ? 

In the notation of formula (54), page 33, v = 100, and it is required 
to find v. From Table 4, for the given salinity and temperature, 

M = 4,644 X10~ 8 , a 2 =-0.774xl0~ 8 

From (54), ^^ = 4,644 X 10~ 8 X 20-0.774 X 10~ 8 x20 2 
° = 10~ 8 X 92,570. 

Therefore, v -v = 0.09257, and v = v - 0.09257 = 99.90743 c. c— 
Answer. 

Example 2. — Under the conditions supposed in ex. 1, the density 
of the water at atmospheric pressure is found to be 1.02375. What 
will be its density under the additional 20 atmospheres pressure ? 

Since density varies inversely as volume, the new density will be 

1 .02375 X 99 ^ 743 = 1.0246983- 

The increase in density, p-p , is 1.0246983-1.02375 = 0.0009486. 
The increase in density may also be found directly from formula 
(55), page — , without bringing in the volumes. In (55) p Q = 1.02375 

p - p = Pok> v + W + a 2)v 2 } 

= 1.02375 {4,644 X10~ 8 X 20 + [(4,644 X10- 8 ) 2 - 0.774 X10" 8 ] 20 2 } 



36 U. S. COAST AND GEODETIC SURVEY. 

p — p = 0.0009485, which agrees substantially with the result found 
by the first method. 

Example 3. — What is the correction to the depth for the difference 
of temperature between air and water due to the change in volume of 
the water with temperature ? (Table 9 gives only the effect of tem- 
perature on the volume of air.) 

The first-mentioned correction is needed only in a table in which the 
water is brought to the surface for measurement. (See p. 18.) 
Take, for example, air temperature, 24° C. water temperature, 10° C. 
salinity, 36, and depth, 27 fathoms. In the notation of formula (27), 
page 19, for 24°, ^ = 1.02442, and for 10°, 5 2 = 1.02775. From 
Table 8, 

b7i\vj 

Therefore, the required correction 

.i i«, 1.02775-1.02442 A , 1flK v 0.00333 

= A 2 h = - 165 X - — 1 Q2442 or A z h== ~ 165 x ~T024~ = 

fathom. 

This example shows the necessity of applying the correction in 
accurate work, as the difference in temperature is by no means 
extreme, and, for a given difference of temperature, the correction 
varies directly as 



(?) 



&/aY 

bh\vj 

which increases rapidly with the depth. 

Example 4- — Suppose that the expansion of water due to the release 
of pressure on coming to the surface has not been allowed for in com- 
puting the scale. (The allowance has been made in Table 8.) What 
correction must be applied to the heights read from the scale; that is, 
what is Aji formula (26), page 19 ? 

Take depth and salinity as in example 3. For depth 27 fathoms, 
p 2 = 4.90 atmospheres (Table 6) . For 10° and salinity 36,m<> = 4474 X 10" 8 , 
and, as before, 

1 = - 165. 



b7i\vj 

Therefore, A 2 ft= -165x4,474 XlO~ 8 X 4.90= -0.036 fathoms. 

At this depth, the correction is small, but A 2 h increases even more 
rapidly with the depth than Aji, so that toward the end of the table 
A 2 7i should not be neglected in accurate work. 



SCALE OF A SOUNDING TUBE. 37 

Table 1. — Miscellaneous physical data compiled from various sourc* 

1. Composition of m:\ water. 

The salinity of sea water is defined as tho number of prams of salts contained in l.OOO prams of sea wafer. 
The relative proportions of i ho various salts in sea water is almost consl mi the world over, except undei 

obviously peculiar conditions. The following may bo taken as represent al I vq: 

Amounts of various salts in 1,000 grams oi sea water; salinity, 85. 









Name. 



Common salt 

Magnesium chloride 

Magnesium sulphate 

Calcium sulphate 

Potassium sulphate — 

Magnesium bromide 

Calcium carbonate and traces of other substances. 



Total. 



Chemical symbol. 



NaCl... 

MgCl 2 . 
Mg SO, 
CiiSd,. 
K 2 S0 4 . 
Mg Bro. 
CaC0 3 



Amount. 



35. 00 



Per cent 
of all 

salts. 



Grams. 




27.21 


77. 75 


3.81 


10.88 


1.66 


4.74 


1.26 


3. 60 


.86 


2.47 


.08 


.22 


.12 


.34 



100.00 



Since, in a dilute solution like sea water the various salts are partially dissociated into their ions, it is 
better to give simply the amount of the separate elements and composite ions. 

Amounts of various elements in 1,000 grams of sea water; salinity So. 



Name. 



Chlorine. . . 
Sodium — 
Magnesium 

Calcium 

Potassium . 
Bromine . . . 



Chemical 
symbol. 



CI. 

Na 
Mg 
Ca. 
K. 
Br. 



Amount. 



Grams. 

19.32 

10.72 

1.32 

.42 

.38 

.07 



Name. 



Sulphuric acid ions 

Carbonic acid ions and traces 
other matter. 



Total 



Chemical 
symbol. 



SO4. 
C0 3 . 



Amount. 



Grams. 
2.69 
.08 



35.00 



2. The salinity, the density, and the chlorine coutent of sea water are connected with one another by the 
three following empirical formulas, the last of which is derived from the two preceding: 

S= 0.030+1.805 CI 

ffo = -0.069+1.4708 Cl-0.00157 (Cl)2+O.00O0398 (Cl)3 
<r = -0.093+0.8149 5-0.000482 5 2 +0.0000068 S* 

S represents the salinity, CI the chlorine content in grams of 1,000 grams of sea water, and o©is a quantity 
such that density at 0° C.= I+TatJq 

The limiting case when CI or S is zero is not that of distilled water, but that of natural water, just 
contiguous to water that is barely brackish, and for this extreme case the formulas should not be pressed 
too closely. 

The value of <r may be found from the salinity by interpolation on the second line of Table 2. 

3. Physical constants of sea water dependent on the salinity. 



Salinity. 


5 


10 


15 


20 


25 


30 


35 


40 


Boiling point, pressure 760 
mm 


100.08°C. 

-0.27°C 

2.93°C 
1. 00415 
0.982 

0. 00138 

1. 33405 


100.16°C 

-0.53 °C 

1.86°C 
1. 00818 
0.968 

0. 00137 

1.33502 


100.23°C 

-0.80°C 

0.77°C 
1. 001213 
0.958 

0. 00136 

1. 33598 


100.31°C 

-1.07°C 

-0.31°C 

1. 001607 

0.951 

0. 00135 

1. 33694 


100.39°C 

-1.35°C 

-1.40°C 

1. 002010 

0.945 

0. 00135 

1. 33790 


100.47°C 

-1.63°C 

-2.47°C 

1. 002415 

0.939 

0.00135 

1.338S5 


100.56°C 
-1.91°C 

-3.52°C 

]. 002X22 
0.932 

0. 00134 

1.33981 


100.64°C 


Freezing point, pressure 760 
mm 


— 2.20°C 


Temperature of maximum 
densit v 


-4.54°C 


Maximum, density 1 


1. 003232 


Specific heat at 17.5°C 

Thermal conductivity (C. G. 
S. units) at 17.5°C 


0.926 
0. 00134 


Index of refraction, D-line at 
18° C 


1. 34077 







1 For the higher salinities the maximum density is attained only by undercooled water. 

Surface tension in dynes per cm.= 77.09— 0.1788f +0.0221 S. 

n ^ ■ * f ■ .* ,. . i , . *• x • n „ « ■* 0.0180(1 +0.00180S -0.00000952) 
Coefficient of viscosity (internal friction) m C. G. S. units= .... rQ^'t+Q 000175^1 ' 

In these equations S is the salinity and t is the temperature centigrade. 



38 



U. S. COAST AND GEODETIC SURVEY. 



4. Physical constants of air. 

Atmospheric air is a mixture containing about 77 per cent nitrogen, 21 per cent oxygen, less than 1 per 
cent of argon and allied* rare gases, and about 1 per cent, on the average, of water vapor. Carbon dioxide 
is only 0.03 per cent. 

One liter of air, temperature 0° C, pressure 1 atmosphere, weighs 1.2928 grams. 

Coefficient of viscosity (C. G. S. units)= (173.3+0.460X10-8. 

Thermal conductivity (C. G. S. units), 0.0000568 (l+O.OOlQOf). 

Specific heat at constant pressure, 0.2377. 

Ratio, specific heat at constant pressure to specific heat at constant volume= 1.405. 

5. Acceleration of gravity at sea level in latitude <p. 

(cm. per sec. per sec.) > 
£=978.039 (1+0.005294 sin 2 ^.-0.000007 sin*2<p) 
=980.621 (1-0.002640 cos 2 ^+0.000007 cos2 2<p) , 

The coefficients of the parentheses in the two forms of the expression for gravity are, respectively, gravity 
at the Equator and at latitude 45°. 



Table 2. — Density of sea water. 

[By salinity is meant the number of grams of salts dissolved in 1,000 grams of sea water. For basis of 
table, see Table 1. Density of pure water at 4° C. is unity, so that tabulated density is weight in grams 
of a cubic centimeter of sea water.) 



^^-^ Salinity 
Temp., °cN. 


6 


8 


10 


12 


14 


16 


18 


20 


22 


—2 


1. 00466 
1. 00478 
1. 00484 
1. 00483 
1. 00476 
1.00463 

1.00445 
1. 00422 
1. 00394 
1. 00361 
1. 00324 

1.00283 
1. 00237 
1. 00188 
1. 00135 
1. 00078 
1.00018 


1.00629 
1. 00640 
1. 00644 
1. 00642 
1. 00634 
1. 00620 

1.00601 
1. 00577 
1. 00548 
1. 00514 
1.00476 

1.00434 
1. 00388 
1. 00338 
1. 00285 
1. 00227 
1.00166 


1. 00792 
1. 00801 
1. 00804 
1. 00801 
1.00791 
1.00776 

1.00756 
1. 00731 
1. 00701 
1. 00667 
1.00628 

1.00586 
1. 00539 
1. 00488 
1. 00434 
1. 00376 
1.00315 


1. 00955 
1. 00963 
1. 00964. 
1. 00959 
1. 00949 
1.00933 

1.00912 
1. 00886 
1. 00855 
1. 00820 
1.00780 

1.00737 
1.00690 
1. 00639 
1. 00584 
1. 00525 
1. 00463 


1.01117 
1.01124 
1.01124 
1.01118 
1. 01106 
1.01089 

1.01067 
1. 01040 
1. 01008 
1. 00973 
1.00932 

1.00888 
1. 00840 
1. 00789 
1. 00733 
1. 00674 
1.00612 


1.01280 
1.01285 
1.01284 
1. 01276 
1.01263 
1.01245 

1.01222 
1.01194 
1.01162 
1.01125 
1.01084 

1.01039 
1. 00991 
1. 00938 
1. 00883 
1. 00823 
1.00760 


1.01442 
1.01446 
1. 01443 
1. 01435 
1.01421 
1.01401 

1.01377 
1. 01348 
1.01315 
1. 01278 
1.01236 

1.01190 
1.01141 
1. 01088 
1. 01032 
1. 00972 
1.00908 


1. 01604 
1. 01607 
1.01603 
1. 01593 
1. 01578 
1.01557 

1.01532 
1.01502 
1. 01468 
1. 01430 
1.01388 

1.01342 
1.01292 
1.01238 
1.01181 
1.01121 
1.01057 


1.01767 





1.01767 


2 


1.01762 


4 


1.01751 


6 


1.01735 


8 


1.01713 


10 


1. 01687 


12 


1. 01657 


14 


1. 01622 


16 


1.01583 


18 


1.01540 


20 


1.01493 


22 


1.01442 


24 


1.01388 


26 


1.01331 


28 


1.01270 


30 


1.01205 







^ s " > \^ Salinity 
Temp.,°C/"\^^ 


24 


26 


28 


30 


32 


34 


36 


38 


40 


—2 


1.01929 
1. 01928 
1. 01922 
1. 01909 
1.01892 
1.01869 

1.01842 
1.01810 
1.01775 
1.01735 
1.01692 

1.01644 
1. 01593 
1.01538 
1. 01480 
1.01419 
1. 01354 


1. 02091 
1. 02089 
1. 02081 
1.02068 
1.02049 
1.02026 

1. 01998 
1.01965 
1. 01929 
1. 01888 
1.01844 

1.01795 
1.01744 
1. 01689 
1. 01630 
1. 01568 
1. 01503 


1.02253 
1. 02250 
1. 02240 
1. 02226 
1. 02206 
1.02182 

1.02153 
1.02119 
1. 02082 
1. 02041 
1.01996 

1.01947 
1.01895 
1.01839 
1.01780 
1.01717 
1. 01652 


1. 02415 
1. 02410 
1. 02400 
1.02384 
1.02364 
1.02338 

1.02308 
1. 02274 
1. 02236 
1. 02194 
1.02148 

1.02099 
1. 02046 
1.01989 
1. 01930 
1. 01867 
1.01801 


1. 02577 
1. 02571 
1.02560 
1. 02543 
1. 02521 
1.02495 

1.02464 
1. 02429 
1. 02390 
1. 02347 
1.02300 

1.02250 
1. 02197 
1.02140 
1. 02080 
1. 02017 
1. 01950 


1. 02739 
1. 02732 
1. 02720 
1. 02702 
1. 02679 
1.02651 

1.02619 
1. 02584 
1. 02544 
1. 02500 
1.02453 

1.02402 
1. 02348 
1. 02291 
1.02230 
1. 02167 
1. 02100 


1. 02902 
1. 02894 
1. 02880 
1.02861 
1. 02837 
1.02808 

1.02775 
1. 02739 
1. 02698 
1. 02654 
1.02606 

1. 02554 
1. 02500 
1. 02442 
1.02381 
1.02317 
1. 02250 


1. 03065 
1. 03055 
1. 03040 
1. 03020 
1. 02995 
1.02965 

1.02932 
1. 02894 
1. 02853 
1. 02807 
1. 02759 

1. 02707 
1. 02652 
1. 02594 
1. 02532 
1. 02468 
1.02400 


1. 03227 





1. 03217 


2 


1.03200 


4 


1.03179 


6 


1. 03153 


8 


1.03123 


10 


1. 03088 


12 


1. 03050 


14 


1. 03007 


16 


1. 02962 


18 


1.02912 


20 


1.02860 


22 


1. 02804 


24 


1. 02745 


26 


1. 02683 


28 


1. 02619 


30 


1. 02551 







1 (Spec. pub. No. 40, U. S. Coast and Geodetic Survey.) 



SCALE OF A SOUNDING TUBE. 



39 



'. 



Table 3. — Vapor pressure of sea water for various salinities and U mperalures. 

The vapor pressures are givon in millimeters of mercury. For basis ol table see p. 23.) 






Temp., ° C 



Salinity. 



— 1. 
0. 
1. 
2. 
3. 
4. 

5. 
6. 
7. 
8. 
9. 

10. 
11. 
12. 
13. 
14. 

15. 
16. 
17. 
18. 
19. 

20. 
21. 
22. 
23. 
24. 



25. 
26. 
27. 
28. 
29. 
30. 



-1. 
0. 
1. 
2. 
3. 
4. 

5. 
6. 

7. 



10. 
11. 
12. 
13. 
14. 

15. 
16. 
17. 
18. 
19. 

20. 
21. 
22. 
23. 
24. 



25. 
26. 
27. 
28. 
29. 
30. 



4.24. 
4.56 
4.91 
5.28 
5.67 
6.08 

6.52 
6.99 
7.49 
8.02 
8.58 

9.18 

9.81 

10.48 

11.20 

11.95 

12.75 
13. 59 
14.49 
15.43 
16.43 

17.48 
18.59 
19.77 
21.00 
22.31 

23.68 
25.13 
26.66 
28.26 
29.95 
31.73 



4.24 
4.56 
4.90 
5.27 
5.66 
6.07 

6.51 
6.98 
7.48 
8.01 
8.57 

9.17 

9.80 
10.47 
11.18 
11.94 

12.73 
13.58 
14.47 
15.41 
16.41 

17.46 
18.57 
19.74 
20.98 
22.28 

23.66 
25.11 
26.63 
28.23 
29.92 
31.69 



in 



4.23 
4.55 
4.90 
5.27 
5.65 
6.07 

6.51 
6.98 
7.47 
8.00 
8.56 

9.16 

9.79 

10.46 

11.17 

11.92 

12.72 
13.56 
14.45 
15.39 
16.39 

17.44 

18.55 
19.72 
20.96 
22.26 

23.63 
25.08 
26.60 
28.20 
29.89 
31.66 



12 



It 



4.23 
4.55 
4.89 
5. 26 
5.65 
6.06 

6.50 
6.97 
7.46 
7.99 
8.55 

9.15 

9.78 

10.45 

11.16 

11.91 

12.71 
13. 55 
14.44 
15.38 
16.37 

17.42 
18.53 
19.70 
20.93 
22.23 

23.61 

25.05 
26.57 
28.17 
29.85 
31.62 



4.22 
4.54 
4.89 
5.25 
5.64 
6.05 

6.49 
6. 96 
7.46 
7.98 
8.54 

9.14 

9.77 

10.44 

11.15 

11.90 

12.70 
13. 53 
14.42 
15.36 
16.35 

17.40 
18.51 
19.68 
20.91 
22.21 

23.58 
25.02 
26.54 
28.14 
29.82 
31.59 



16 



4.22 
4.54 
4.88 
5. 25 
5.64 
6.05 

6.49 
6.95 

7.45 
7.97 
8.53 

9.13 

9.76 

10.43 

10.14 

11.88 

12.68 
13. 52 
14.40 
15.34 
16.34 

17.38 
18.49 
19.66 
20.89 
22.18 

23.55 
24.99 
26.51 
28.11 
29.78 
31.55 



is 


20 


4.21 


4.21 


4.53 


4.r>:\ 


4.88 


4.87 


5.34 


5.24 


5. 63 


5.62 


6.04 


6.03 


6.48 


6. 17 


6.94 


(i.'.ll 


7.44 


7.43 


7.97 


7.96 


8.52 


8.51 


9.12 


9.11 


9.75 


9.74 


10.41 


10.40 


11.12 


11.11 


11.87 


11 86 


12.66 


12.65 


13. 50 


13.49 


14.39 


14.37 


15. 33 


15. 31 


16.32 


16.30 


17.36 


17.34 


18.47 


18.45 


19. 63 


19. 61 


20.86 


20.84 


22.16 


22.13 


23.53 


23.50 


24.97 


24.94 


26.48 


26. 45 


28.07 


28.04 


29.75 


29.72 


31.52 


31.48 



Temp.,°C 



Salinity. 



24 



4.20 
4.52 
4.86 
5.22 
5.61 
6.02 

6.46 
6.92 
7.41 
7.94 
8.49 

9.09 

9.71 

10.38 

11.08 

11.83 

12.62 
13.45 
14.34 
15.27 
16.26 

17.30 
18.41 
19.57 
20.79 
22.08 

23.45 
24.88 
26.39 
27.98 
29.65 
31.41 



26 



4.19 
4.51 
4.85 
5.22 
5.60 
6.01 

6.45 
6.91 

7.40 
7.93 
8.48 

9.08 

9.70 

10.37 

11.07 

11.82 

12.60 
13.44 
14.32 
15.26 
16.24 

17.29 
18.38 
19.54 
20.77 
22.06 

23.42 
24.-85 
26.36 
27.95 
29.62 
31.37 



28 



4.19 
4.51 
4.85 
5.21 
5.60 
6.01 

6.44 
6.90 
7.40 
7.92 
8.47 

9.07 

9.69 

10.35 

11.06 

11.80 

12.59 
13.42 
14.31 
15.24 
16.22 

17.27 
18.36 
19.52 
20.74 
22.03 

23.39 
24.82 
26.33 
27.92 
29.58 
31.34 



30 



4.18 
4.50 
4.84 
5.21 
5.59 
6.00 

6.43 
6.90 
7.39 
7.91 
8.46 

9.06 

9.68 

10.34 

11.04 

11.79 

12.58 
13.41 
14.29 
15.22 
16.20 

17.25 
18.34 
19.50 
20.72 
22.01 

23.36 
24.79 
26.30 
27.88 
29.55 
31.30 



32 



4.18 
4.50 
4.84 
5.20 
5.58 
5.99 

6.43 
6.89 
7.38 
7.90 
8.45 

9.05 

9.67 

10.33 

11.08 

11.77 

12.56 
13.39 
14.27 
15.20 
16.19 

17.22 
18.32 
19.48 
20.70 
21.98 

23.34 
24.77 
26.27 
27.85 
29.52 
31.26 



34 



4.17 
4.49 
4.83 
5.19 
5.58 
5.98 

6.42 

6.88 
7.37 
7.89 
8.44 

9.03 

9.66 

10.32 

11.02 

11.76 

12.55 
13.78 
14.26 
15.18 
16.17 

17.20 
18.30 
19.45 
20.67 
21.96 

23.31 
24.74 
26.24 
27.82 
29.48 
31.23 



36 



4.17 
4.49 
4.83 
5.19 
5.57 
5.98 

6.41 

6.87 
7.36 
7.88 
8.43 

9.02 

9.65 

10.31 

11.01 

11.75 

12. .53 
13.36 
14.24 
15.17 
16.15 

17.18 
18.28 
19.43 
20.65 
21.93 

23.28 
24.71 
26.21 
27.78 
29.44 
31.19 



38 



4.16 
4.48 
4.82 
5.18 
5.56 
5.97 

6.40 
6.86 
7.35 

7.87 
8.42 

9.01 

9.63 

10.29 

10.99 

11.73 

12. 52 
13. 35 
14.22 
15. 15 
16.13 

17.16 
18.26 
19.41 
20.62 
21.90 

23.25 
24.68 
26.17 
27.75 
29.41 
31.15 



4.20 
4.52 
4.87 

5.23 
5. 62 
6.03 

6.46 
6.93 

7.42 
7 95 
8.50 

9.10 

9.72 

10.39 

11.10 

11.84 

12.63 
13.47 
14.36 
15.29 
16.28 

17.32 
18.43 

19. 59 
20.82 
22.11 

23.47 
24.91 
26.42 
2S.01 
29.68 
31.44 



40 



4.16 
4.48 
4.82 
5.17 
5. .56 
5.96 

6.40 
6.86 
7.34 
7.86 
8.41 

9.00 

9.62 

10.28 

10.98 

11.72 

12. .50 
13.33 
14.20 
15.13 
16.11 

17.14 
18.23 
19.38 
20.60 
21.88 

23.23 
24.65 
26.14 
27.72 
29.37 
31.12 



40 



U. S. COAST AND GEODETIC SURVEY. 



Table 4. — Compressibility of sea watpr. 1 

[Table 4a gives IO'Xm, m« being the true coefficient of compressibility at atmospheric pressure, for water 
of various salinities and temperatures. Table 4b gives IC^X^, «2 being the coefficient of the second 
term in the formula (55), p. 33 for the change in relative volume, for water of various salinities and 
temperatures. The unit of pressure in both tables is the atmosphere. Temperatures are in degrees 
centigrade.] 

Table 4a.— 10 8 Xm - 



— - __ Salinity. 

Temp., ° C. ~~~~~— ________ 


5 


10 


15 


20 


25 


30 


35 


40 





5,106 
4,942 
4,808 
4,701 
4,617 
4,554 
4,509 


5,037 

4,880 
4,751 
4,648 
4,568 
4,508 
4,465 


4,970 
4,819 
4,695 
4,596 
4,520 
4,462 
4,421 


4,905 
4,759 
4,640 
4,546 
4,473 
4,418 
4,378 


4,842 
4,701 
4,587 
4,497 
4,426 
4,374 
4,335 


4,779 
4,644 
4,535 
4,448 
4,381 
4,330 
4,293 


4,719 
4,589 
4,484 
4,400 
4,336 
4,287 
4,251 


4 659 


5 


4* 534 


10 


4* 434 


15 


4' 354 


20 


4 292 


25 


4' 245 


30 


4' 210 







Table 4b.— 10 8 Xa 2 



Temp., ° C. 



Salinity. 



5 


10 


15 


20 


25 


30 


35 


-0. 897 


-0. 880 


-0. 864 


-0. 849 


-0. 834 


-0. 820 


-0. 806 


.841 


.826 


.812 


.799 


.786 


.774 


.762 


.796 


.783 


.771 


.759 


.747 


.736 


.725 


.761 


.749 


.738 


.727 


.717 


.707 


.697 


.737 


.726 


.716 


.706 


.696 


.686 


.676 


.723 


.713 


.703 


.693 


.683 


.674 


.664 


-0. 720 


-0. 710 


-0. 700 


-0. 690 


-0. 680 


-0. 670 


-0.660 



40 



0. 
5. 
10 
15 
20 
25 
30 



-0.793 
.750 
.715 

.687 

.667 

.655 

-0. 650 



Table 5. — Absorption of atmospheric gases by sea water. 

[The tabular values are the number of cubic centimeters of gas that can be absorbed by sea water of the 
given salinity at the given temperature. The volumes are reduced to a temperature of 0°C., and a pressure 
of one atmosphere. Carbon dioxide (C0 2 ) is absorbed rather freely by sea water, but, since the total vapor 
pressure of C0 2 is only about 0.0003 atmosphere, its absorption need not be considered in connection with 
he sounding tube.] ' • 



1. OXYGEN. 



~~ — -——___ Salinity. 
Temp., ° C. -— . _^__^ 





5 


10 


15 


20 


25 


30 


35 


40 


—2 


c.c. 
10.88 
10.29 
9.03 
8.02 
7.22 
6.57 
6.04 
5.57 


c.c. 

10.53 
9.97 
8.75 
7.79 
7.03 
6.40 
5.88 
5.42 


c.c. 

10.18 
9.65 
8.48 
7.56 
6.83 
6.22 
5.72 
5.27 


c.c. 
9.84 
9.33 
8.21 
7.33 
6.63 
6.05 
5.56 
5.12 


c.c. 
9.50 
9.01 
7.94 
7.10 
6.43 
5.88 
5.40 
4.96 


c.c. 
9.16 
8. 68 
7.67 
6.87 
6.23 
5.70 
5.24 
4.80 


c.c. 
8.82 
8.36 
7.40 
6.63 
6.04 
5.53 
5.08 
4.65 


c.c. 
8.47 
8.03 
7.13 
6.40 
5.84 
5.35 
4.93 
4.50 


c.c. 
8.12 





7.71 


5 


6.86 


10 


6.17 


15 


5.64 


20 


5.18 


25 


4.77 


30 


4.35 







2. NITROGEN. 



—2 


19.45 
18.56 
16. 60 
14.97 
13.63 
12.54 
11.66 
10.94 


18.83 
17.97 
16.10 
14.55 
13.27 
12.24 
11.40 
10.70 


18.18 
17.37 
15. CO 
14.13 
12.91 
11.93 
11.13 
10.46 


17.61 
16.77 
15. 10 
13.70 
12.55 
11.63 
10.86 
10.22 


16.90 
16.18 
14.59 
13.27 
12.20 
11.32 
10.59 
9.98 


16.27 
15.58 
14.09 
12.85 
11.84 
11.02 
10.32 
9.74 


15.63 
14.99 
13.59 
12.43 
11.48 
10.71 
10.05 
9.50 


15.00 
14.40 
13.08 
12.00 
11.12 
10.40 
9.78 
9.26 


14.36 





13.80 


5 


12.58 


10 


11.57 


15 


10.76 


20 


10.09 


25 


9.51 


30 


9.02 







1 See note 4, p. 32. 



SCALE OF A SOUNDING TUBE. 



41 



Table 6. — Pressure of sea water at various depths, byJormula(85) pag< 

[Density 1.025; temp. 10°C; gravity as in lat. 45°.] 



Depth, h (fathoms). 


Water 
pressure, 

V». 
(atmos- 
pheres). 


Depth, h (fathoms). 


\\';iler 
pressure, 

Vi 

(atmos- 
pheres). 


1 >epth, h ( fathoms). 


Water 
pressure, 

Pa 

(atmos- 
pheres). 


1 


0. 18142 
.36284 
. 54426 
. 72569 

0.90712 

1. 08855 
1.26997 
1.45141 
1.63284 
1. 81427 

1.99571 
2. 17714 
2. 35858 
2. 54002 
2.72147 

2.90291 
3.08436 
3. 26580 

3. 44725 
3. 62870 

3.81015 
3.99160 
4. 17306 
4. 35452 

4. 53597 

4. 71743 
4. 89889 
5.08036 
5. 26182 

5. 44329 

5. 62475 
5. 80622 
5. 98769 
6. 16916 
6. 35064 

6. 53211 

6. 71359 
6. 89506 
7. 07654 

7. 25803 


41 


7. 43951 
7. 62099 
7. 80248 
7. 98397 
8. 16545 

8. 34694 

8. 52844 
8. 70993 
8. 89142 
9. 07292 

9. 25442 
9. 43592 
9. 61742 
9. 79892 
9. 98043 

10. 16193 
10. 34344 
10. 52495 

10. 70646 
10. 88797 

11. 06949 

11. 25100 
11. 43252 
11.61404 

11. 79556 

11.97708 
12. 15860 
12. 34013 

12. 52166 

12. 70318 

12. 88471 
13. 06624 

13. 24778 

13. 42931 
13.61084 

13. 79238 
13. 97392 
13. 15546 
13. 33700 

14. 51855 


81 


1 1.70009 


2 


42 


82. . 


14. smc, i 


3 


43... 


83. 


15. 06310 


4 


44... 


84. 


15. 23544 


5 


45... 


85... 


15. 42629 


6 


46 


86. . 


15. 60784 




47 


87... 


15. 78939 


8 


48 


ss. 


15. 97095 


9... 


49 


89. . 


16. 15251 


10... 


50... 


90... 


16. 33407 


11 


51 


91 


16. 51563 


12 


52 


92... 


16.69719 


13 


53 


93 


16. 87875 


14... 


54... 


94. 


17. 06032 


15... 


55 


95. . 


17. 24189 


16 


56 


96 


17. 42345 


17 


57 


97 


17. 60442 


18. 


58. . . 


98. 


17. 78660 


19... 


59 


99... 


17. 96817 


20... 


60 


100... 


18. 14974 


21 


61 


101 


18. 33132 


22 


62 


102 


18. 51290 


23 


63 


103... 


18.69448 


24 


64 


104 


18. 87606 


25 


65 


105... 


19. 05764 


26 


66 


106 


19. 23922 


27 


67 


107 


19.42081 


28 


68 


108 


19. 60240 


29 


69 


109 


19. 78698 


30... 


70 


110. 


19. 96558 


31 


71 


Ill 


20. 14717 


32 


72 


112 


20. 32876 


33 


73 


113 


20. 51035 


34 


74 


114 


20.69195 


35 


75 


115 


20. 87355 


36 


76 


116 


21. 05515 


37 


77 


117 


21. 23675 


38 


78 


118 


21. 41835 


39 


79 


119 


21. 59996 


40... 


80... 


120... 


21. 78156 











42 U. S. COAST AND GEODETIC SURVEY. 

Table 7. — Table for scale of sounding tube. 
[Temperature of air, 15° C; of water, 10°; humidity of air, 100%; surface density of water, 1.025; gravity as 

in latitude 45 . 1 — is volume of air when volume of tube is taken as unity. 



than three significant figures and is always greater than unity.] 






is given to not more 





Volume 
of air 
space 

Vi 

Vi 


<t) 


i-5 

Vi 


Depth, ft (fathoms). 


Volume 
of air 
space 

V2 

V\ 


»(f) 


1-^ 


Depth, ft (fathoms). 


dh 
(Units of 

fifth 
decimal 
place). 


dft 
(Units of 

fifth 
decimal 
place). 


"5ft(§-) 


i>h\ vj 


1 


0. 82630 
. 71529 
. 63057 
. 56376 
.50978 

. 46521 
.42781 
. 39596 
. 36853 
.34465 

.32367 
. 30510 

.28854 
. 27369 
. 26029 

.24813 
. 23706 
. 22694 
.21764 
. 20908 

.20116 
. 19382 
. 18699 
.18063 
. 17469 

. 16912 
.16390 
. 15899 
. 15436 
. 15000 

. 14587 
. 14197 
. 13827 
. 13476 
. 13142 

. 12824 
. 12521 
. 12232 
. 11956 
. 11692 

.11440 
.11198 
.10966 
.x0744 
.10530 

. 10325 

.10128 
. 09938 
. 09755 
.09578 

.09408 
.09244 
.09085 
.08931 
.08783 

.08640 
. 08501 
.08367 
.08236 
. 08110 






61 


0.07980 
. 07869 
. 07753 
.07641 
.07532 

. 07426 
. 07323 
. 07223 
. 07126 
.07031 

. 06939 
.06849 
. 06761 
. 06676 
.06593 

.06511 
.06432 
.06355 
. 06279 
. 06205 

. 06133 
.06063 
. 05994 
. 05927 
. 05861 

.05796 
. 05733 
. 05672 
.05611 
.05552 

.05494 
.05438 
.05382 
. 05328 
. 05275 

.05223 
. 05171 
. 05121 
. 05072 
.05023 

.04976 
. 04929 
.04884 
. 04839 
. 04795 

.04752 
. 04709 
. 04667 
.04626 
.04586 

.04547 
. 04508 
. 04469 
.04432 
. 04395 

. 04359 
.04223 
. 04288 
.04253 
. 04219 


- 121 

- 117 

- 114 

- 110 

- 107 

- 104 

- 102 

- 99 

- 96 

- 94 

- 91 

- 89 

- 87 

- 84 

- 82 

- 80 

- 78 

- 76 

- 75 

- 73 

- 71 

- 70 

- 68 

- 67 

- 65 

- 64 

- 62 

- 61 

- 60 

- 58 

- 57 

- 56 

- 55 

- 54 

- 53 

- 52 

- 51 

- 50 

- 49 

- 48 

- 47 

- 46 

- 45 

- 44 

- 44 

- 43 

- 42 

- 41 

- 41 

- 40 

- 39 

- 39 

- 38 

- 37 

- 37 

- 36 

— 36 

- 35 

- 34 

— 34 


- 763 


2 






62 


- 787 


3 






63 


- 811 


4 






64 


- 836 


5 






65 


- 862 


6 






66 


- 887 


7 






67 


— 913 


8 






68 


- 940 


9 






69 


- 966 


10 


-2233 

-1970 
-1751 
-1566 
-1409 
-1274 

-1158 
-1057 

- 969 

- 892 

- 823 

- 762 

- 707 

- 658 

- 614 

- 575 

- 539 

- 506 

- 476 

- 449 

- 424 

- 401 

- 380 

- 360 

- 342 

- 326 

- 310 

- 296 

- 282 

- 270 

- 258 

- 247 

- 237 

- 227 

- 218 

- 209 

- 201 

- 194 

- 187 
• - 180 

- 173 

- 167 

- 161 

- 156 

- 151 

- 146 

- 141 

- 137 

- 132 

- 128 

- 124 


- 29 

- 34 

- 40 

- 45 

- 52 

- 58 

- 65 

- 72 

- 80 

- 88 

- 96 

- 105 

- 114 

- 124 

- 133 

- 144 

- 154 

- 165 

- 177 

- 188 

- 200 

- 213 

- 226 

- 239 

- 253 

- 267 

- 281 

- 296 

- 311 

- 326 

- 342 

- 359 

- 375 

- 392 

- 409 

- 427 

- 445 

- 464 

- 483 

- 502 

- 522 

- 542 

- 562 

- 583 

- 604 

- 626 

- 648 

- 670 

- 693 

- 716 

- 739 


70 


— 994 


11 


71 


-1020 


12 


72 


-1050 


13 


73 


-1080 


14 


74 


-1110 


15 


75 


-1140 


16 


76 


-1160 


17 


77 


-1190 


18 


78 


-1220 


19 


79 


-1250 


20 


80 

81..... 


-1280 


21.. 


-1310 


22 : 


82 


-1350 


23 


83 


-1380 


24 


84 


-1410 


25 


85 


-1450 


26 


86 


-1480 


27 


87 


-1510 


28 


88 


-1550 


29 


89 


-1580 


30 


90 


-1610 


31 


91 


-1650 


32 


92 

93 


-1690 


33 


-1730 


34 


94 


-1760 


35 


95 


-1790 


36 


96 


-i830 


37 


97 


-1870 


38 


98 


-1900 


39 


99 


-1940 


40 


100 


-1980 


41 


101 


-2020 


42 


102 


-2060 


43 


103 


-2100 


44 


104 


-2140 


45 


105 


-2180 


46 


106 


-2220 


47 


107 


-2260 


48 


108... 


—2310 


49 


109 


—2350 


50 


110 


—2390 


51 


Ill 


—2430 


52 


112... 


—2470 


53 


113 


—2520 


54 


114.. 


—2560 


55 


115 


—2610 


56 


116 


—2650 


57 


117... 


—2690 


58 


118 . 


—2730 


59 


119. 


—2780 


60 


120 


—2840 









i See formulas (7) to (11a), pp. 9 and 10, and (21) and following, pp. 18-20 



SCALE OF A SOUNDING TUBE. 



43 



Table 8. — Special I able for scale of Coast and Geodetic Survey tube. 

[Temperature of air, 60° F., of water, 50° F. = 10° C; humidity of air, 100%; surf: 



* i 



1.025; gravity as inlat.45V — is volume of air when volume of tube is taken as unit v. — — ; — r-is given 
Vi d / i_A 

oh \7J 
to not more than three significant figures and is always creator t ban unit v 

The last three columns give the length of rod of the diameter stated thai tnu I be inserted in order to till 
the air space, thus bringing the water to the top of the tube. The numbers are computed for a tube 24 inches 
long and A inch in diameter.] 



Depth, h (fathoms). 


Volume of 
air space 


dli V'i/ 

units of 
fifth deci- 
mal place. 


1-^ 


Length of rod to bring water 
»p (inches). 


J inch 
diameter. 


A inch 
diametei . 


finch 
diameter. 


1 


0. 97561 
. 82455 
. 71399 
.62955 
. 56287 

.50911 
. 40465 
. 42733 
. 39554 
.36815 

.34431 
.32330 
.30481 

. 28827 
. 27343 

. 26004 
. 24790 
.23683 
. 22671 
.21741 

. 20885 
. 20093 
. 19359 
. 18676 
. 18040 

. 17445 
. 16888 
.16366 
. 15874 
. 15411 

. 14974 
. 14561 
. 14170 
. 13800 
. 13448 

.13113 
. 12795 
. 12491 
. 12202 
. 11925 

. 11661 
. 11408 
.11165 
. 10933 
. 10709 

. 10495 
. 10289 
. 10091 
. 09900 
. 09716 

.09539 

.09368 
.09203 
.09044 
. 08890 












2 












3 












4 












5 










24.016 


6 










21.722 


7 










19 s''.") 


8 








• 


18. 232 


9 








24.302 

22. 019 

21. 15 J 
19.868 

18. 728 
17.711 
16. 799 

15. 977 
15. 231 
14. 551 
13. 929 
13.358 

12.832 
12. 345 
11. 894 
11.475 
11.084 

10. 718 

10. 376 

10. 055 

9.753 

9.469 

9.200 
8.946 
8.700 
■ 8. 478 
8.262 

8.057 
7.861 
7. 675 
7.497 
7.327 

7.164 
7.010 
6.860 
6.717 
6.580 

6.448 
6.322 
6.200 
6.083 
5.970 

5.861 
5. 756 
5.654 
5.556 
5.462 


16 876 


10 


-2549 

-2229 
-1967 
-1749 
-1565 
-1408 

-1273 

-1158 
-1057 

- 969 

- 892 

- 823 

- 762 

- 708 

- 659 

- 615 

- 575 

- 539 

- 506 

- 477 

- 450 

- 425 

- 402 

- 381 

- 361 

- 343 

- 326 

- 311 

- 296 

- 383 

- 270 

- 259 

- 248 

- 237 

- 228 

- 219 

- 210 

- 202 

- 194 

- 187 

- 180 

- 174 

- 168 

- 162 

- 157 

- 151 


- 25 

- 29 

- 34 

- 40 

- 45 

- 52 

- 58 

- 65 

- 72 

- 80 

- 88 

- 96 

- 105 

- 114 

- 123 

- 133 

- 143 

- 154 

- 165 

- 176 

- 188 

- 200 

- 212 

- 225 

- 239 

- 252 

- 266 

- 280 

- 295 

- 310 

- 326 

- 341 

- 357 

- 374 

- 391 

- 408 

- 426 

- 444 

- 462 

- 481 

- 500 

- 520 

- 539 

- 559 

- 579 

- 601 




15. 70s 


11 




14 691 


12 




13. 797 


13 




13.005 


14 




12.300 


15 




11.666 


16 


24.964 
23.798 
22. 736 
21.764 
20. 872 

20.049 
19. 289 
18. 584 
17.929 
17.318 

16. 747 
16. 213 
15.711 
15. 239 
14. 795 

14.375 
13. 979 
13. 603 
13.247 
12. 910 

12. 589 
12.283 
11.992 
11.713 
11.448 

11.194 
10. 951 
10. 718 
10. 495 
10.281 

10. 075 
9.878 
9. 687 
9.504 
9.328 

9.158 
8.994 
8.835 
8.682 
8.534 


11.095 


17 


10. 577 


18 


10. 105 


19 


9. 673 


20 


9.276 


21 


8.911 


22 


8.573 


23 


8 260 


24 


7.969 


25 


7.697 


26 


7.443 


27 


7.206 


28 


6.983 


29 


6.773 


30 


6.575 


21 


6 389 


32 


6.213 


33 


6 046 


34 


5.888 


35 


5.738 


36 


5.595 


37 


5 459 


38 


5 330 


39 


5 206 


40 


5.088 


41 


4 975 


32 


4 867 


43 


4 764 


44 


4 664 


45 


4.569 


46 


4 478 


47 


4 390 


48 


4.305 


49 


4 224 


50 


4 146 


51 


4 070 


52 


3 997 


53 


3 927 


54 


3 859 


55 


3.793 



i See formulas (7) to (lib), pp. 9 and 10, and remark on p. 19. 



44 U. S. COAST AND GEODETIC SURVEY. 

Table 8. — Special table for scale of Coast and Geodetic Survey tube — Continued. 





Volume of 
air space 

02. 
Vl 


dhyvxJ 
units of 
fifth deci- 
mal place. 


1 v 2 

b /v 2 \ 

dh\vi) 


Length of rod to bring water 
to top (inches). 


Depth, h (fathoms). 


Jineh 
diameter. 


■rVinch 
diameter. 


finch 
diameter. 


56 


0. 08741 
. 08596 
.08457 
.08322 
.08191 

. 08063 
.07940 
. 07821 
. 07704 
.07592 

.07482 
.07375 
. 07272 
.07171 
.07073 

.06977 

. 06884 
.06793 
.06705 
.06618 

.06534 
.06452 
.06372 
.06294 
.06218 

.06143 
.06070 
.05999 
.05929 
.05861 

.05794 
.05729 
. 05666 
.05603 
.05542 

. 05482 
.05424 
.05366 
.05310 
.05255 

. 05200 
.05147 
. 05095 
. 05044 
.04994 

. 04945 
. 04897 
. 04850 
.04803 
.04757 

.04713 
. 04669 
. 04625 
.04583 
.04541 

.04500 
. 04459 
. 04420 
. 04381 
.04342 

. 04304 
.04267 
.04231 
. 04195 
.04159 


- 147 

- 142 

- 137 

- 133 

- 129 

- 125 

- 121 

- 118 

- 114 

- Ill 

- 108 

- 105 

- 102 

- 99 

- 97 

- 94 

- 92 

- 90 

- 87 

- 85 

- 83 

- 81 

- 79 

- 77 

- 76 

- 74 

- 72 

- 70 

- 69 

- 67 

- 66 

- 64 

- 63 

- 62 

- 60 

- 59 

- 58 

- 57 

- 56 

- 55 

- 54 

- 53 

- 52 

- 51 

- 50 

- 49 

- 48 

- 47 

- 46 

- 45 

- 44 

- 44 

- 43 

- 42 

- 41 

- 41 

- 40 

- 39 

- 39 

- 38 

- 37 

- 37 

- 36 

- 36 

- 35 


- 622 

- 644 

- 666 

- 688 

- 711 

- 734 

- 758 

- 782 

- 806 

- 830 

- 855 

- 881 

- 906 

- 932 

- 959 

- 986 
-1010 
-1040 
-1070 
-1100 

-1130 
-1150 
-1180 
-1210 
-1240 

-1270 
-1300 
-1330 
-1360 
-1400 

-1430 
-1460 
-1490 
-1530 
-1560 

-1590 
-1630 
-1660 
-1700 
-1730 

-1770 
-1800 
-1840 
-1880 
-1910 

-1950 
-1990 
-2030 
. -2070 
-2100 

-2140 
-2180 
-2220 
-2260 
-2300 

-2340 
-2380 
-2430 
-2470 
-2510 

-2560 
-2600 
-2640 
-2680 
-2720 


8.391 
8.253 
• 8. 119 
7.989 
7.863 

7.741 
7.623 
7.508 
7.396 

7.288 

7.183 
7.080 
6.981 
6.884 
6.790 

6.698 
6.608 
6.521 
6.436 
6.354 

6.273 
6.194 
6.117 
6.042 
5.969 

5.897 
5.827 
5. 759 
5.692 
5.627 

5.563 
5.500 
5.439 
5.379 
5.320 

5.263 
5.207 
5.151 
5.097 
5.044 

4.992 
4.941 
4.892 
4.843 
4.794 

4.747 
4.701 
4.656 
4.611 
4.567 

4.524 
4.482 
4.440 
4.399 
4.359 

4.320 
4.281 
4.243 
4.206 
4.169 

4.132 
4.096 
4.061 
4.027 
3.993 


5.370 
5.282 
5.196 
5.113 
5.032 

4.954 
4.878 
4.805 
4.734 
4.664 

4.597 
4.531 
4.468 
4.406 
4.345 

4.287 
4.230 
4.174 
4.119 
4.066 

4.015 
3.964 
3.915 
3.867 
3.820 

3.774 
3.729 
3.686 
3.643 
3.601 

3.560 
3.520 
3.481 
3.443 
3.405 

3.368 
3.332 
3.297 
3.262 
3.228 

3.195 
3.163 
3.131 
3.099 
3.068 

3.038 
3.009 
2.980 
2.951 
2.923 

2.895 
2.868 
2.842 
2.816 
2.790 

2.765 
2.740 
2.716 
2.692 
2.668 

2.645 
2.622 
2.599 
2.577 
2.555 


3.729 


57 


3.668 


58 


3.609 


59 


3.551 


60 


3.495 


61 


3.440 


62 


3.388 


63 


3.337 


64 


3.287 


65 


3.239 


66 


3.192 


67 


3.147 


68 


3.103 


69 


3.060 


70 


3.018 


71 


2.977 


72 


2.937 


73 


2.898 


74 


2.861 


75 


2.824 


76 


2.788 


77 


2.753 


78 


2.719 


79 


2.685 


80 


2.653 


81 


2.621 


82 


2.590 


83 


2.560 


84 


2.530 


85 : 


2.501 


86 


2.472 


87 


2.444 


88 


2.417 


89 


2.391 


90 


2.365 


91 


2.339 


92 


2.314 


93 


2.290 


94 


2.266 


95 


2.242 


96 


2.219 


97 


2.196 


98 


2.174 


99 


2.152 


100 


2.131 


101 


2.110 


102 


2.089 


103 


2.069 


104 


2.049 


105 


2.030 


106 


2.011 


107 


1.992 


108 


1.973 


109 


1.955 


110 


1.937 


Ill 


1.920 


112 


1.903 


113 


1.886 


114 


1.869 


115 


1.853 


116 


1.837 


117 


1.821 


118 


1.805 


119 


1.790 


120 


1.775 







Sl'ALl'L OK A SOUNDING TUBE. 



45 



Table 1>. — Corrections to sounding tube readings for U mperatun and pressure. 

[Computed for air temperature^ 60°F.= 154° (V, water temperature 50° f.= io° c. Barometer— 30 in. 

70.200 cm. See equation (15), p. 12.] 

1. CORRECTION FOR TEMPERATURE. 



Depth from scale, h 
(fathoms). 



5.. 
10. 
20. 
30. 
■10. 
50. 
60. 
70. 
SO. 
90. 
100 
110 



Temperature of air minus temperature of water ( Fahrenheil I. 



—10° 



Fath- 
oms. 
+ 1.0 
+ 1.5 
+ 2.5 
+ 3.4 
+ 4.4 
+ 5.4 
+ 6.4 
+ 7.3 
+ 8.3 
+ 9.3 
+ 10.3 
+ 11.2 



-30° 



Fath- 
oms. 
+0.8 
+ 1.2 
+ 2.0 
+2.8 
+3.5 
+4.3 
+5.1 
+5.9 
+6.6 
+ 7.4 
+8.2 
+9.0 



•20° 



Fath- 
oms. 
+0.6 
+ .9 

+ 1.5 
+2.1 
+ 2.6 
+3.2 
+3.8 
+4.4 
+5.0 
+5.6 
+6.2 
+6.7 



■10 c 



Fath- 
oms. 
+ 0.4 
+ .6 
+ 1.0 
+ 1.4 
+ 1.8 
+2.2 
+2.6 
+2.9 
+3.3 
+3.7 
+ 4.1 
+4.5 



0° +10° +20° +30° +10° +50 



Fath- 
oms. 

+0.2 
+ .3 
+ .5 

+ .7 
+ .9 
+ M 
+ 1.3 
+ 1.5 
+ 1.7 
+ 1.9 
+ 2.0 
+2.2 



Fath- 
oms. 
0.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 



Fath- 
oms. 
-0.2 

- .3 

- .5 

- .7 

- .9 
-1.1 
-1.3 
-1.5 
-1.7 
-1.9 
-2.0 
-2.2 



Fath- 
oms. 

-0. ! 

- .6 
-1.0 
-1.4 
-1.8 
-2.2 
-2.6 
-2.9 
-3.3 
-3.7 
-4.1 
-4.5 



Fath- 
oms. 

-0.6 
- .9 
-1.5 
-2.1 
-2.6 
-3.2 
-3.8 
-4.4 
-5.0 
-5.6 
-6. 2 
-6.7 



Fath- 
oms. 

-1.2 
-2.0 
-2.8 
-3.5 
-4.3 
-5.1 
-5.9 
-6.6 
-7.4 
-S.2 
-9.0 



+ 00° 



Fath- 
oms. 

- 1.0 

- 1.5 

- 2.5 
-3.4 

- 4.4 
-5.4 

- 6.4 

- 7.3 
-8.3 
-9.3 
-10.3 
-11.2 



2. CORRECTION FOR PRESSURE. 



Depth from scale, h 
(fathoms). 



10. 
20. 
30. 
40. 
50. 
60. 
70. 
80. 
90. 
100 
110 



Barometer reading in inches. 



29.0 



Fath- 
oms. 
-0.3 
- .7 
-1.0 
-1.3 
-1.7 
-2.0 
-2.3 
-2.7 
-3.0 
-3.3 
-3.7 



29.2 



Fath- 
oms. 
-0.3 

- .5 

- .8 
-1.1 
-1.3 
-1.6 
—1.9 
-2.1 
-2.4 
-2.7 
-2.9 



29.4 29.6 29.8 30.0 30.2 30.4 



Fath- 
oms. 
-0.2 

- .4 

- .6 

- .8 
-1.0 
-1.2 
-1.4 
-1.6 
-1.8 
-2.0 
-2.2 



Fath- 
oms. 
-0.1 

- .3 

- .4 

- .5 

- .7 

- .8 

- .9 
-1.1 
-1.2 
-1.3 
-1.5 



Fath 
oms. 
-0.1 

- .1 

- .2 

- .3 

- .3 

- .4 

- . 5 

- .5 

- .6 

- .7 

- .7 



Fath- 
oms. 
0.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 
.0 



Fath- 
oms. 
+0.1 
+ .1 



+ .5 
+ .5 

+ .6 
+ .7 

+ .7 



Fath- 
oms. 
+0.1 
+ .3 
+ .4 
+ .5 
+ .7 
+ .8 
+ .9 
+ 1.1 
+ 1.2 
+ 1.3 
+ 1.5 



30.6 



Fath- 
oms. 
+0.2 
+ .4 
+ .6 
+ .8 
+ 1.0 
+ 1.2 
+ 1.4 
+ 1.6 
+ 1.8 
+ 2.0 
+2.2 



30.8 



Fath- 
oms. 
+0.3 
+ .5 
+ .8 
+ 1.1 
+ 1.3 
+1.6 
+ 1.9 
+2.1 
+2.4 
+2.7 
+2.9 



31.0 



Fath- 
oms. 
+0.3 
+ .7 
+ 1.0 
+ 1.3 
+ 1.7 
+2.0 
+2.3 
+2.7 
+3.0 
+3.3 
+3.7 



Table 10. — Effect of absorption of air on scale of sounding tube. 

[Column 1 shows the depth in fathoms. Column 2 shows the relative volume of air at the given depth. 
Column 2 has been taken from Table 7, and column 3 shows the relative volume of air if the sea 
water in the tube were saturated with atmospheric gases. 2 Column 4 shows the correction to be applied 
to the depths read from a scale computed on the supposition of no absorption, when, in point of fact, absorp- 
tion had gone on to the saturation point.] 



Depth (fathoms). 



5. 
10 
15 
20 
25 
30 
35 
40 
45 
50 



V2IV1 
No ab- 
sorption. 



0. 5098 
.3447 
.2603 
.2091 
.1747 
.1500 
.1314 
.1169 
.1053 
.0958 



V2IV1 
Complete 
satura- 
tion. 



0. 5052 
.3363 
.2496 
.1968 
.1613 
.1358 
.1166 
.1016 
.0896 
.0797 



Correc- 
tion to 
depth 
(fath- 
oms). 



0.07 
.3 
.8 

1.4 

2.1 

3.0 

4.0 

5.1 

6.3 

7.7 



Depth (fathoms). 



55. 
60. 
65. 
70. 
75. 
80. 
85. 
90. 
95. 
100 



V2IV1 
No ab- 
sorption. 



0. 0878 
.0811 
.0753 
.0703 
.0659 
.0621 
.0586 
.0555 
.0528 
.0502 



V2IV1 
Complete 
satura- 
tion. 



0.0714 
.0644 
.0584 

.0533 
. 04S7 
.0447 
.0411 
.0379 
.03,50 
.0324 



Correc- 
tion to 
depth 
(fath- 
oms). 



— 9.2 
-10.8 
-12.4 
-14.2 
-16.1 
-18.1 
-20.1 
-22.2 
-24.5 
-26.7 



1 See equation (7), p. 9. t?=lQ° C; h=15° C; pi+p'i=l atmosphere. s See equation (20), p. 16. 



O 



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